# Full text of "The Elements of Euclid; Sir Thomas Heath-2nd Ed. ebox Set"

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THE THIRTEEN BOOKS OF EUCLID'S ELEMENTS THE THIRTEEN BOOKS OF EUCLID'S ELEMENTS TRANSLATED FROM THE TEXT OF HEIBERG WITH INTRODUCTION AND COMMENTARY Sir THOMAS L. HEATH, K.C.B., K.C.V.O., F.R.S., SC.D. CAMS., HON. D.SC. OXFORD HONORARY FELLOW (SOMETIME FELLOW) OF TRINITY COLLEGE CAMBRIDGE SECOND EDITION REVISED WITH ADDITIONS VOLUME n BOOKS III— IX DOVER PUBLICATIONS, INC. NEW YORK J ^A'JV^' < ]^[ >Mi-j i ii I ' r, I I 'III I , Thl( new edlHon, Uttt publlihed in 1956, U an un&b ridged and unaltered lepublicatton of tJie lecond edition. It It publlthed through ipeclal airoogement with Cambridge Unlvcnity Preti. Librajy of Congress Catalog Card Numbers ^6-4)36 Manufactured in the United States of Amerioi Dover Publicaciona, Inc. 180 Varick Street ■^ New York 14, N. Y. CONTENTS OF VOLUME 11. Book III. Definitions I Propositions ..... 6 Book IV. Definitions . . 78 Propositions 80 Book V. Introductory note 1 1 z Definitions U3 * Propositions • ■ 138 Book VI. Introductory note . . 187 Definitions 188 Propositions i9r Book VII. Definitions «77 Propositions 296 Book VIII. , 345 Book IX. 384 Greek Index to Vol. II 417 English Index to Vol. II 431 BOOK III. DEFINITIONS. [. £qual circles are those the diameters of which are equal, or the radii of which are equal. 2. A straight iine ts said to touch a circle which, meeting the circle and being produced, does not cut the circle, 3. Circles are said to touch one another which, meeting one another, do not cut one another. 4. In a circle straight lines are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal. 5. And that straight line is said to be at a greater distance on which the greater perpendicular falls. 6. A segment of a circle is the figure contained by a straight line and a circumference of a circle. 7. An angle of a segment is that contained by a straight line and a circumference of a circle. 8. An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined. mi . . .. 9. And, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference, » BOOK in [hi. deff. 10. A sector of a circle is the figure which, when an angle is constructed at the centre of the circle, is contained by the straight lines containing the angle and the circumference cut off by them. 11. Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another. Definition i. Iiroi kvkXkh turivi wy at Stdfitrpoi urai tltrtVj ^ mv al fK rur HtrTptnv ivai tUrir, Many editors have held that this should not have been included among deAnitions. Some, e.g. Tartaglia, would call it apos/u/aU; others, e.g. Borelli and Playfair, would c^l it an axiom ; others again, as Billingsley and Clavius, while admitting it as a definitien, add explanations based on the mode of constructing a circle ; Simson and Pfleiderer hold that it is a tfuoretn, I think however that Euclid would have maintained that it is a definition in the proper sense of the term ; and certainly it satisfies Aristotle's requirement that a "definitional statement" (opurTdtM AoyM) should not only state the fait (to iri) but should indicate the cause as well {De aitima ii. i, 413 a 13). The equality ot circles with equal radii can of course be proved by superposition, but, as we have seen, Euclid avoided this method wherever he could, and there is nothing technically wrong in saying " By equal circles 1 mean circles with equal radii." No flaw is thereby introduced into the system of the Elements ; for the definition could only be objected to if it could be proved that the equality predicated of the two circles in the definition was not the same thing as the equality predicated of other equal figures in the Elements on the l^is of the Congruence- Axiom, and, nt^less to say, this cannot be proved because it is not true. The existence of equal circles (in the sense of the definition) follows from the existence of equal straight tines and I. Post. 3, The Greeks had no distinct word for radius, which is with them, as here, the {straight line drawn) from the centre 7 Jk r™ mVrpou ((Wiln) ; and so definitely was the expression appropriated to the radius that in tov Kit^pou was used without the article as a predicate, just as if it were one word. Thus, eg., in III. I JK KtvTpov yap means " for they are radii " : cf, Archimedes, On the Sphere and Cylinder i\. z, ij BE Ik rm xiirpoa iarl Tm,,,KiK^jm, BM is it radius of the circle. Definition 2. Euclid's phraseology here shows the regular distinction between ainvr$vii and its compound li^maOai, the former meaning "to tnett" and the latter "to touch." The distinction was generally observed, by Greek geometers from Euclid onwards. There are however exceptions so far as hrrfaBm is concerned; thus it means "to touch" in Eucl. iv. Def 5 and sometimes in Archimedes On the other band, ^c/iairTccrdat is used by Aristotle in certain III. DEFF. i— 4] DEFINITIONS 3 cases where the orthodox geometrical term would be airrifrtfot. Thus in Meleerohgica m. 5 (376 b 9) he says a certain circle will pass through all the angles (ajTotrulK i^nu^tTM t^v yioi'tiui'), atid (376 a 6) M will lie on a given (circular) circumference {iihofian)^ trtpti^tptiiK c^'i^crat to M). We shall find awrviBai used in these senses in Book iv. Deff. 2, 6 and Deff. 1, 3 respectively. The latter of the two expressions (quoted from Aristotle means that thi locus of M is a given drde, just as in Pappus o^trai to armiiov Oia-n StSo/tiyrft ti$nat means that th^ locus ^the point is a straight line given in position. Definition 3. Todhunter remarks that different opinions have been held as to what is, or should be, included in this definition, one opinion being that it only means that the circles do not cut in the neighbourhood of the point of contact, and that it must be shown that they do not cut elsewhere, while another opinion is that the definition means that the circles do not cut at all Todhunter thinks the latter opinion correct. I do not think this is proved ; and I prefer to read the definition as meaning simply that the circles meet at a point but do not cut at that point. I think this interpretation preferable for the reason that, although Euclid does practically assume in III. ti — :3, without stating, the theorem that circles touching at one point do not intersect anywhere else, he has given us, before reaching that point in the Book, means for proving for ourselves the truth of that statement. In particular, he has given us the propositions in. 7, 8 which, taken as a whole, give us more information as to the general nature of a circle than any other propositions that have preceded, and which can be used, as will be seen in the sequel, to solve any doubts arising out of Euclid's unproved assumptions. Now, as a matter of fact, the propositions are not used in any of the genuine proofs of the theorems in Book in. ; in. 8 is required for the second proof of ni. 9 which Simson selected in preference to the first proof, but the first proof only is regarded by Heibecg as genuine. Hence it would not be easy to account for the appearance of in. 7, 8 at all unless as affording means of answering possible objections {cf. Proclus' explanation of Euclid's reason for inserting the second part of i. 5). External and internal contact are not distinguished in Euclid until 111. II, 12, though the^^w of in. 6 (not the enunciation in the original text) represents the case of internal contact only. But the definition of touching circles here given must be taken to imply so much about internal and external contact respectively as that (a) a circle touching another internally must, immediately before " meeting " it, have passed through points within the circle that it touches, and {b) a circle touching another externally must, immediately before meeting it, have passed through points outside the circle which it touches. These facts must indeed be admitted if internal and external are to have any meaning at all in this connexion, and they constitute a minimum admission necessary to the proof of in. 6. Definition 4. 'Ev kukA^ urof Ltck-^w airo roC Kci'Tpou cf^tuii AryotTui, orat' at a.vh TOv Ktyrpov iv aitras Kti0tTQi oiyOfm^t i(7u4 ciKTti^. BOOK III [in. DEFF. 5—9 Definition s- Definition 6. T/t^/ia KuicXov hm to wtpit^ofifyoy (rj^^/ia iro n (v^ttat Hat kvkXov - .. Definition 7. TfHj/iOTOS 8c yMwla tirrlv jj ircpit)(onfirr) v'lni t* iJS«w »«ii KUtkov wtfyi^ptlat, ThU definition is only interesting historically. The an^& of a segment, being the " angle " formed by a straight line and a " circumference," is of the kind described by Proclus as " mixed." A particular " angle " of this sort is the "angle of a semicircle," which we meet with again in ui. i6, along with the so-called "horn -like angle" (jMparmtSijs), the supposed "angle" between a tangent to a circle and the circle itself. The " angle of a semicircle " occurs once in Pappus (vii. p. 670, 19}, but tt there means scarcely more than the corner of a semicircle regarded as a point to which a straight line is directed. Heron does not give the definition of the att£k of a segment, and we may conclude that the mention of it and of the angle of a umieircle in Euclid is a survival from earlier text-books rather than an indication that Euclid considered either to be of importance in elementary geometry (cf. the note on iii- i6 below). We have however, in the note on i. s above (Vol. 1, ppi 252—3), seen evi- detice that the a»^& ^aj<^Mf had played some part in geometrical proofs up to Euclid's time. It would appear from the passage of Aristotle there quoted {Anal, prior, i, 24, 41 b 13 sqq.) that the theorem of 1. 5 was, in the text-books immediately preceding Euclid, proved by means of the equality of the two " angles of" any one segment. This latter property must therefore have been regarded as more elementary (for whatever reason) than the theorem of i. 5 ; indeed the definition as given by Euclid practically implies the same thing, since it speaks of only one " angle of a segment," namely "/At angle contained by a straight line and a circumference of a circle," Euclid abandoned the actual use of the "angle" in question, but no douht thought it unnecessary to bieak with tradition so far as to strike the definition out also. Definition 8. tnffutoy nal air a^oi> ittI Tft ir^ara tt^^ tvOtiti^, 7f itrrt ^ovif TOtJ TfiT/jfj^aro^f Definition 9. 'Orav Bt at vepiij(owTat r^v yuiviav eo&tiai dirakafi^tii'tiMri xim irtpn^tpiiav, III. DErr. it^ ii] NOTES ON DEFINITIONS 5—11 J Definition 10. To/uvt Si icvK\m/ itrriv, Smv irpof ry ithirp^ rov NtLncXou murra^ yaria, TO n*pttx^firvotf <r)mfia into re Tmv TTfr ytaviav vtptt^ov^r^v (u^ttwv tcai n^ djroXa/A^tLvoftivjjt vtt* airrSv vtpu^*p<itK. A scholiast says that it was the shoemaket's knife, trtarrvrti^uA^ roiitit, which su^ested the name ra^t for a. sector of a circle. The derivation of the name from a resemblance of shape is parallel to the u^ of ap^irXos (also a sMetmakfr't knife) to denote the well known figure of the Book of Lemmas partly attributed to Archimedes. A wider definition of a sector than that given by Euclid b found in a Greek scholiast (Heiberg's Euclid, Vol. v. p. 260) and in an-Nairizi (ed. Curtze, p. hi). "There are two varieties of sectors ; the one kind have the angular vertices at the centres, the other at the circumferences. Those others which have their vertices neither at the circumferences nor at the centres, but at some other points, are for that reason not called sectors but sector-like figures (td/m«(8ij v-x^partk)," The exact agreement between the scholiast and an-NairizI suggests that Heron was the authority for this explanation. The Mctor-Hkt figure bounded by an arc of a circle and two lines drawn from its extremities to meet at any point actually appears in Euclid's book On divisions {trtpl Sinifiwtuiv) discovered in an Arabic MS. and edited by Woepcke (cf. Vol. 1. pp. 8—10 above). This treatise, alluded to by Proclus, had for its object the division of figures such as triangles, trapezia, ([uadrilaterals and circles, by means of straight lines, into parts equal or in given ratios. One proposition e.g. is, Ta divide a triangle into two equal parts by a straight lint passing through a given point on one side. The proposition (28) in which the quasi-udor occurs is, To divide suth a figure by a straight line into two equal parts. The solution in this case is given by Cantor (Gesck d. Math, u, pp. aS;— 8). If ABCD be the given figure, E the middle point of BD and EC at right angles to BD, the broken line AEC clearly divides the figure into two equal parts. Join AC, and draw EF parallel to it meeting AB\n F. Join CF, when it is seen that CF divides the figure into two equal parts. Definition u, ' - De Morgan remarks that the use of the word similar in "similar segments " is an anticipation, and that similarity of form is meant. He adds that the definition is a theorem, or would be if " similar " had taken its final meaning. BOOK III. PROPOSITIONS. Proposition i. To find the centre of a given circle. Let ABC be the given circle ; thus it is required to find the centre of the circle ABC. Let a straight line AB be drawn s through it at random, and let it be bisected at the point D ; from D let DC be drawn at right angles to AB and let it be drawn through to E ; let CE be bisected at F\ "o I say that F is the centre of the circle ABC. For suppose it is not, but, if possible, let G be the centre, and let GA, GD, GB be joined. ,., , .. , IS Then, since AD is equal to DB, and DG is common, the two sides AD, DG are equal to the two sides BD, DG respectively ; and the base GA is equal to the base GB, for they are 20 radii ; therefore the angle ADG is equal to the angle GDB. [i. 8] But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right ; [i. Def, 10] 35 therefore the angle GDB is right. 111. i] PROPOSITION I 7 But the angle FDB is also right ; therefore the angle FDB is equal to the angle GDB, the greater to the less : which is impossible. Therefore G is not the centre of the circle ABC. 30 Similarly we can prove that neither is any other point except F. Therefore the point F is the centre of the circle ABC. , PoRiSM. From this it is manifest that, if in a circle a straight line cut a straight line into two equal parts and at 35 right angles, the centre of the circle is on the cutting straight line. Q. E. F. a. For auppose It is not. Tliis is expressetl in the Greek by the two wocds H^i yif, but Biicb an EllLpticaJ phrase is impossible in English. 17. the two sides AD, DG are etjual to the two aides BD, DO respectively. As before observed, Euclid 13 not always oarerul to put the equals in correspond Itig order. The (est here has " CZ>, D3." 'Fodhunter observes that, when, in the construction, DC is said to be productd to J?, it is assumed that D is within the circle, a fact which Euclid first demonstrates in in. 2. This is no doubt true, although the word iai^^w, " let it be drawn through^' is used instead of iK^ijiKi^trSm, " let it be^rcJuad." And, although it is not necessary to assume that I> is within the circle, it is necessary for the success of the construction that the straiglit line drawn through jD at right angles to AB shall meet the circle ir_ two points (and no more): an assumption which we are not entitled to make on the basis of what has gone before only. Hence there is much to be said for the alternative procedure recommended by De Morgan as preferable to that of Euclid. De Morgan would first prove the fundamental theorem that "the line which bisects a chord perpendicularly must contain the centre," and then make ni. i, iii. 25 and iv. 5 immediate corollaries of it. The fundamental theorem is a direct consequence of the theorem that, if P is any point equidistant from A and .5, then P lies on the straight line bisecting AJ3 perpendicularly. We then take any two chords AB, j4Cof the given circle and draw £>0, EO bisecting them perpendicularly. Unless BA^ AC are in one straight line, the straight lines DO, EO must meet in some f)oint O (see note on iv. 5 for possible methods of proving this). And, since both DO, EO must contain the centre, must be the centre. This method, which seems now to be generally preferred to Euclid's, has the advantage of showing that, in order to find the centre of a circle, it is sufficient to know three points on the circumference. If therefore two circles have three points in common, they must have the same centre and radius, so that two circles cannot have three points in common without coinciding entirely. Also, as indicated by De Morgan, the same construction enables us (i) to draw the complete circle of which a segment or arc only is given {ill. 25), and (2) to circumscribe a circle to any triangle (iv. 5). 8 BOOK. Ill [ill. I. 1 But, if the Greeks had used this construction for finding the centre of a circle, they would have considered it necessary to add a proof that no other point than that obtained by the construction can be the centre, as is clear both from the similar rtduetio ad abturdum in iii i and also from the fact that Euclid thinks it necessary to prove as a separate theorem (ui. 9) that, if a point within a circle be such that three straight lines (at least) drawn from it to the circumference are equal, that point must be the centre. In fact, honrever, the proof amounts to no more than the remark that the two perpendicular bisectors can have no more than one point common. And even in De Morgan's method there is a yet unproved assumption. In order that DO, EO may meet, it is necessary that AB, AC should not be in one straight line or, in other words, that BC should not pass through A. This results from iii. 2, which therefore, stKctly speaking, should precede. To return to Euclid's own proposition HI. i, it will be observed that the demonstration only shows that the centre of the circle cannot lie on either side of CD, so that it must lie on CD or CD produced. It is however taken for granted rather than pioved that the centre must be the middle point of CE. The proof of this by rtduetio ad absurdum is however so obvious as to be scarcely worth giving. The same consideration which would prove it may be used to show that a circle cannot have more than one ctntre, a proposition which, if thought necessary, may be added to iii. i as a corollary. Simson o^rved that the proof of [ii. i could not but be by reductio ad aksurdum. At the beginning of Book in. we have nothing more to base the proof upon than the dejinitton of a circle, and this cannot be made use of unless we assume some point to be the centre. We cannot however assume that the point found by the construction is the centre, because that is the thing to be proved. Nothing is therefore left to us but to assume that some other point is the centre and then to prove that, whatever other point is taken, an absurdity results; whence we can infer that the point found is the centre. The Porism to in. i is inserted, as usual, parenthetically before the words Svtp I5» T«7<7at, which of course refer to the problem itself. Proposition 2. If on the circumference of a circle two points be taken at random, the straight line joining the points will fall within the circle. Let ABC be a circle, and let two points A, B \x. taken at random on its circumference ; . I say that the straight line joined from .^ to Z? will fall within the circle. For stippose it does not, but, if possible, let it fall outside, as AEB ; let the centre of the circle ABC be taken [in. 4 and let it be Z? ; let DA, DB be joined, and let DFE be drawn through. III. i] PROPOSITIONS I, J 9 • Then, since DA is equal to DB, the angle DAE is also equal to the angle DBE. [i. 5] And, since one side AEB of the triangle DAE is produced, the angle DEB is greater than the angle DAE. [i. 16] But the angle DAE is equal to the angle DBE ; therefore the angle DEB is greater than the angle DBE. And the greater angle is subtended by the greater side ; [i. 19] therefore DB is greater than DE. But DB is equal to DF\ ' ~ - therefore DF is greater than DE, the less than the greater : which is impossible. Therefore the straight Une joined from A to ^ will not fall outside the circle. Similarly we can prove that neither will it fall on the circumference itself; therefore it will fall within. ,< , Therefore etc. • • Q. E. D. The nduitio ad absurdum form of proof is not really necessary in this case, and it has the additional disadvantage that it requires the destruction of two hypotheses, namely (hat the chord is (i) outside, (i) on ihe circle. To prove the proposition directly, we have only to show that, if ^ be any point on the straight line AB between A and B, DE is less than the radius of the circle. This may be done by the method shown above, under i. 24, for proving what is assumed in that proposition, namely that, in the hgurc of the proposition, /"falJs beiow EG if DE is not greater than DF. The assumption amounts to the following proposition, which De Morgan would make to precede 1. ^4 ; " Every straight line drawn from the vertex of a triangle to the base is less than the greater of the two sides, or than either if they be equal." The case here Is that in which the two sides are equal ; and, since the angle DAB is equal to the angle DBA, while the exterior angle DEA is greater than the interior and opposite angle DBA, it follows that the angle DEA is greater than the angle DAE, whence DE must be less than DA or DB. Camerer points out that we may add to this proposition the further statement that all points on AB produced in either direction are outside the circle. This follows from the proposition (also proved by means of the theorems that the exterior angle of a triangle ts greater than either of the interior and opposite angles and that the greater angle is subtended by the greater side) which De Morgan proposes to introduce after i. 3 1, namely, " The perpendicular is the shortest straight line that can be drawn from a 10 BOOK in [ill. 2, 3 given p^oint to a given straight line, and of others that which is nearer to the perpendicular is less than the more lemote, and the converse ; also not more than two equaJ straight lines can be drawn from the point to the line, one on each side of the perpendicular." The fact that not more than two equal straight lines can be drawn from a given point to a given straight line not passing through it is proved by Proclus on ], 1 6 (see the note to that proposition) and can alternatively be proved by means of i. 7, as shown above in the note on I. 1 2. It follows that A straight line cannot cut a circle in Men than two points a proposition which De Morgan would introduce here after in. a. The proof given does not apply to a straight line passing through the centre j but that !iiich a line only cuts the circle in two points is self evident Proposition 3. If in a circle a straight line through the centre bisect a straight line not through the centre, it also cuts it at right angles : and if it cut it at right angles, it also bisects it. Let ABC be a circle, and in it let a straight line CD J throi]gh the centre bisect a straight line AB not through the centre at the point F; I say that it also cuts it at right angles. For let the centre of the circle ABC 10 be taken, and let it be .£"; let EA, EB be joined. Then, since AF is equal to FB, and FE is common, two sides are equal to two sides ; IS and the base EA is equal to the base EB ; therefore the angle AFE is equal to the angle BFE. [1.8] Biit, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right ; [i- Def. 10] 20 therefore each of the angles AFE, BFE is right. Therefore CD, which is through the centre, and bisects AB which is not through the centre, also cuts it at right angles. Again, let CD cut AB at right angles ; «5 I say that it also bisects it. that is, that AFis equal to FB. in. 3. 4] PROPOSITIONS 2—4 11 For, with the same construction, since £A is equal to EB, the angle EAF is also equal to the angle EBF. [1. 5] But the right angle AFE is equal to the right angle BEE, JO therefore EAF, EBF are two triangles having two angles equal to two angles and one side equal to one side, namely EF, which is common to them, and subtends one of the equal angles ; therefore they will also have the remaining sides equal to 35 the remaining sides ; [i- 26] therefore AF is equal to FB. Therefore etc. Q. E. D. 46. with the eame construction^ Ttav ntW-up jrnracrjrFLacr^^rrup . This proposition asserts the two partial converses (cf, note on i. 6) of the Porism to iti. i. De Morgan would place it next to in. i. pROPOSITiON 4. If in a circle two straight lines cut one another which are not through the centre, they do not bisect one another. Let A BCD be a circle, and In it let the two straight lines AC, BD, which are not through the centre, cut one another at E \ I say that they do not bisect one another. For, if possible, let them bisect one another, so that AE is equal to EC, and BE to ED ; let the centre of the circle ABCD be taken [in, 1], and let it h^ F\ let FE be joined. Then, since a straight line FE through the centre bisects a straight line AC not through the centre, it also cuts it at right angles ; [ni, 3] therefore the angle FEA is right. Again, since a straight line FE bisects a straight line BD, it also cuts it at right angles ; fm. 3] therefore the angle FEB is right. xa BOOK III [ill. 4, 5 But the angle FEA was also proved right ; therefore the angle FEA is equal to the angle FEB, the less to the greater : which is impossible. Therefore AC, BD do not bisect one another. Therefore etc. y. K. D. Pkovosition 5. 1/ two circles cut one another, they will not have ike same centre. For let the circles ABC, CDG cut one another at the points B, C\ I say that they will not have the same centre. For, if possible, let it be E \ let EC be joined, and let EFG be drawn through at random. Then, since the point E is the centre of the circle ABC, EC is equal to EF. [i. Def. 15] Again, since the point E is the centre of the circle CDG, EC is equal to EG. But EC was proved equal to iS'/'also ; therefore EF is also equal to EG, the less to the greater : which is impossible. Therefore the point E is not the centre of the circles ABC, CDG. Therefore etc. ' . ' . Q. E. D. The propositions nt. 5, 6 could be combined in one. It makes no difference whether the circles cut, or meet without cutting, so long as they do not coincide altogether; in either case they cannot have the same centre. The two cases are covered by the enunciatiorv ; If the circumferexces of two ctTctti meet at a point th^ {annot have the same centre. On the other hand, If two circles have the same autre and one point in their circumferences common, they mitst coincide altogether. III. 6] PROPOSITIONS 4—6 n Proposition 6. 1/ two cinles touch one another, they will not have the same centre. For let the two circles ABC, CDE touch one another at the point C\ I say that they will not have the same centre. For, if possible, let it be F; let FC be joined, and let FEB be drawn through at random. Then, since the point F is the centre of the circle ABC, FC is equal to FB. Again, since the point F is the centre of the circle CDE, FC is equal to FE. „ But FC was proved equal to FB ; therefore FE is also equal to FB, the less to the greater: which is impossible. Therefore F is not the centre of the circles ABC, CDE. Therefore etc, Q. E, D. The English editions enunciate this propusltion of circles touching inttmaliy, but the word (<vt<k) is a mere interpolation, which was no doutrt made because Euclid's figure showed only the case of internal contact. The fact is that, in his usual manner, he chose for demonstration the more difficult case, and left the other case (that of external contact) to the intelligence of the reader. It is indeed sufficiently self-evident that circles touching extemally cannot have the same centre ; but Euclid's proof can really be used for thia case too. Camerer remarks that the proof of iti. 6 seems to assume tacitly that the points E and B cannot coincide, or that circles which touch internally at C cannot meet in any other point, whereas this fact is not proved by Euclid till jii. 13. But no such general assumption is necessary here; it is only necessary that one Une drawn from the assumed common centre should meet the circles in different points; and the very notion of internal contact requires that, before one circle metis the other on its inner side, it must have passed through points within the latter circle. 1^4 ^ BOOK Ul (|n^«f -•'"•■ Proposition 7. // OK the diameter of a circle a point be taken which is not the centre of the circle, and front the point straight lines fall upon the circle, (hat will be greatest on which the centre is, tlie remainder of the same diameter will be least, and of ike rest 5 the nearer to the straight line through the centre is always greater than the more remote, and only two equal straight lines will fall from the point on the circle, one on each side of ike least straight line. Let ABCD be a circle, and let AD be a diameter of it ; 10 on v4Z) let a point F be taken which is not the centre of the circle, let E be the centre of the circle, and from F let straight lines FB, FC, FG fall upon the circle ABCD; I say that FA is greatest, FD is least, and of the rest FB is IS greater than FC, and FC than FG. For let BE, CE, GE be joined. Then, since in any triangle two sides are greater than the remaining one, [i- 20] ao EB, EF are greater than BF. But AE is equal to BE ; therefore AF\% greater than BF. Again, since BE is equal to CE^ and FE is common, ' ' " ' 25 the two sides BE, EF are equal to the two sides CE, EF. But the angle BEFh also greater than the angle CEF; therefore the base BF is greater than the base CF. [i. 24] For the same reason CF is also greater than FG, , , ,. 30 Again, since GF, FE are greater than EG, and EG is equal to ED, w- - • . GF, FE are greater than ED. Let EE be subtracted from each ; therefore the remainder GF is greater than the remainder i%FD. Therefore FA is greatest, FD is least, and FB is greater than FC, and FC than FG. I III. 7] PROPOSITION 7 *S 1 say also that from the point F only two equal straight lines will fall on the circle A BCD, one on each side of the 40 least FD, For on the straight line EF, and at the point E on it, let the angle ^^//^ be constructed equal to the angle GEF\\- aal. and let FH be joined. Then, since GE is equal to EH, 4i and EF is common, the two sides GE, EF are equal to the two sides HE, EF; and the angle GEF is equal to the angle HEF ; therefore the base FG is equal to the base FH. [i- 4] I say again that another straight line equal to FG will no; so fall on the circle from the point F. For, if possible, let FK so fall. Then, since FK is equal to FG, and FH to FG, ' tJ,ii. • FK is also equal to FH, -^n-i. the nearer to the straight tine through the centre being ss thus equal to the more remote : which is impossible. Therefore another straight line equal to GF m\\ not fall from the point F upon the circle ; therefore only one straight line will so fall. Therefore etc. '' Q. E. D, 4, of the same diameter. I have iriseite<i Ihese words Tot clearness* sake. The text Kas simply ^Xax^ri^ A^ ij X«ir^, " and the remaining (straight line) least." 7, 39. one on CAcb side. The word *' one * is not in the Greek, but is necessary to g:iv< the force of t^' hdrtpa r^ iKaxirnp, literally " on both sides," or " on each nrtlie two sides, of the leajit." De Morgan points out that there is an unproved assumption in this tietnonstration. We draw straight lintjs from F, as FB, FC, such that the angle DFB is greater than the angle DFC and then assume, with respect to the straight lines drawn from the centre E to B, C, that the angle DEB is greater than the angle DEC. This Is most easily pitived, I think, by means of the converse of part of the theorem about the lengths of different straight lines drawn to a given straight tine from an external point which was mentioned above in the note on III. J. This converse would be to the effect that, 1/ two unequal straight lines be drawn from a point to a gitxn straight line whieh are not perpendicular to the straight line, tht greater of the hm is tht further from the perfettdicular from the point to the given straight line. This can either be proved from its converse by rtductio ad absurdum, or established directly by means of i. 47. Thus, in the accompanying figure, FB must cut .£C in some point M, since the angle BFE is less than the angle CFE. Therefore EM is less than EC, and therefore than EB. tuff t« BOOK III [in. 7 Hence the point B in which FB meets the cticie is further from the foot of the petpendicuUr from E on FB than i£'\% ; therefore the angle BEF\% greater than the angle CEF, Another way of enunciating the first part of the proposition is that of Mr H. M. Taylor, viz. " Of all straight lines drawn to a circle from an internal point not the centre, the one which passes through the centre is the greatest, and the one which when produced passes through the centre is the least; and of any two others the one which mbUnds the greater angle at the centre is the greater." The substitution of the angle subtended at the antre as the criterion no doubt has the effect of avoiding the necessity of dealing with the unproved assumption in Euclid's proof referred to above, and the similar substitution in the enunciation of the first part of i[i. 8 has the effect of avoidmg the necessity tor dealing with like unproved assumptions in Euclid's proof, as well as the complication caused by the distinction in Euclid's enunciation between lines falling from an external point on the convex eircumfercnce and on the ccneave dreumfe'rence of a circle respectively, terms which are not defined but taken as understood. Mr Nixon {Euclid Eevised) similarly substitutes as the criterion the angle subtended at the centre, but gives as his reason that the words " nearer " and " more remote " in Euclid's enunciation are scarcely clear enough without some definition of the sense in which they are used, Smith and Bryant make the substitution in iii. 8, but follow Euclid in lii. 7. On the whole, 1 think that Euclid's plan of taking straight lines drawn from the point which is not the centre direct to the circumference and making greater or less angles at that point with the straight line containing it and the centre b the more instructive and useful of the two, since it is such lines drawn in any manner to the circte from the point which are immediately useful in the proofs of later propositions or in resolving difficulties connected with those proofs. Heton again (an-Nairizi, ed, Curtze, pp. 114^5) ^^^ * "o'^ o" ''*'* proposition which is curious. He first of alt says that Euclid proves that lines nearer the centre are greater than those more remote fi'om it. This is a different view of the question from that taken in Euclid's proposition as we have it, in which the lines are not nearer to and more remote from the centre but from the line through the centre. Euclid takes lines inclined to the latter line at a greater or less angle ; Heron introduces distance from the centre in the sense of Deff. 4, 5, i.e. in the sense of the length of the perpendicular drawn to the line from the centre, which Euclid does not use till iii, t4, 15. Heron then obsen'es that in Euclid's proposition the lines compared are all drawn on one side of the line through the centre, and sets himself to prove the same truth of lines on opposite sides which are more or less distant ^iww the centre. The new point of view necessitates a quite different line of proof, anticipating the methods of later propositions. The first case taken by Heron is that of two straight lines such that the perpendiculars from the centre on them fall on the lines themselves and not in either case on the line produced. Let A be the given point, D the centre, and let AE be nearer the centre than AF, so that the perpendicular DG on AE is less than the perpen- dicular DIfoa AF. Then sqs. on DG, G£ = sq%. on DH, HF, and sqs. on DG, GA = sqs. on DH, HA. But sq. on I>G < sq. on DH. III. 7, 8] PROPOSITIONS 7, 8 iJ Therefore sq. on GE > sq. on MJ^l iind sq. on GA > sq. on ffA, whence G£ > J/F, GA^HA. ,.- . Therefore, by addition, AE > AF. The other case taken by Heron is that where one perpendicular fails on the line produced, as in the annexed figure. In this case we prove in like manner that GE > HF, and GA > AH, Thus AE is greater than the sum of HF, AH, whence, a fortiori, AE is greater than the difference of HF, AH, i.e. than AF. Heron does not give the third possible case, that, namely, where both perpendiculars fall on the lines produced, The fact is that, in this case, the foregoing method breaks down. Though AE be nearer to the centre than AF'in the serjse that DG is less than DH, AE is not greater but less than AF. Moreover this cannot be proved by the same method as before. For, while we can prove that GE> HF, GA > AH, we ca.nnot make any inference as to the comparative length of AE, AF. To judge by Heron's corresponding note to in. 8, he would, to prove this case, practically prove iii. 35 first, i.e. prove that, if EA be produced to K and FA to Z, rect. FA, AL = lect. £A, AK, from which he would infer that, since AK^ AL by the first case, AE <AF. An excellent moral can, I think, be drawn from the note of Heroa Having the appearance of supplementing, or giving an alternative for, Euclid's proposition, it cannot be said to do more than confuse the subject. Nor was It necessary to find a new proof for the cast where the two lines which are compared are on epposiit sides of the diameter, since Euclid shows that for each line from the point to the circumference on one side of the diameter there is another of the same length equally inclined to it on the other side. \ Proposition 8. !f a point be taken outside a circle and front the point straight lines be drawn through to the circle, one of which is through the centre and the others are drawn at random, then, of the straight lines which fall on the concave circum- ference, that through the centre is greatest, while of the rest 1» BOOK III [ill. 8 ike nearer to that through the centre is always greater than the more remote, but, of the straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least is always less than the more remote^ and only two equal straight lines will fall on the circle from the point, one on each side of the least. Let ABC be a circle, and let a point D be taken outside ABC; let there be drawn through from it straight lines DA, DE, DF, DC, and let DA be through the centre ; I say that, of the straight lines falling on the concave circumference AEFC, the straight line DA through the centre is greatest, while DE is greater than DF and DF than DC; but, of the straight lines falling on the convex circumference HLKG, the straight line DG between the point and the diameter AG is least; and the nearer to the least DG is always less than the more remote, namely DK than DL, and DL than DH. For let the centre of the circle ABC be taken [m. i], and let xth^M; let ME, MF, MC, MK, ML, MH be joined. Then, since AM is equal to EM, let MD be added to each ; therefore AD is equal to EM, MD. But EM, MD are greater than ED ; [i. lo] therefore AD is also greater than ED. Again, since ME is equal to MF, and MD is common, therefore EM, MD are equal to FM, MD ; and the angle EMD is greater than the angle FMD ; therefore the base ED is greater than the base FD. [..«4] Similarly we can prove that FD is greater than CD ; therefore DA is greatest, while DE is greater than DF, and DF than DC. III. 8] PROPOSITION 8 19 Next, since MK, KD are greater than MD, [i. 30] and MG is equal to MK, therefore the remainder KD is greater than the remainder GD, so that GD is less than KD, And, since on MD, one of the sides of the triangle MLD, two straight lines MK, KD were constructed meeting within the triangle, therefore MK, KD are less than ML, LD \ ., [i. 21] and MK is equal to ML ; therefore the remainder DK is less than the remainder DL. Similarly we can prove that DL is also less than DH ; therefore DG is least, while DK is less than DL, and DL than DH. I say also that only two equal straight lines will fall from the point D on the circle, one on each side of the least DG, On the straight line MD, and at the point M on it, let the angle DMB be constructed equal to the angle KMD, and let DB be joined. Then, since MK is equal to MB, and MD is common, the two sides KM, MD are equal to the two sides BM, MD respectively ; and the angle KMD is equal to the angle BMD ; therefore the base DK is equal to the base DB. [i. 4] I say that no other straight line equal to the straight line DK will fall on the circle from the point D. For, if possible, let a straight line so fall, and let it be DN, Then, since DK is equal to DN, while DK is equal to DB, DB is also equal to DN, that is, the nearer to the least DG equal to the more remote: which was proved impossible. Therefore no more than two equal straight lines will fall on the circle ABC from the point D, one on each side ot ZJ£? the least. Therefore etc. 90 BOOK III [ill. 8 As De Morgan points out, there are here two assumptions similar to that tacitly made in the proof of iii. 7, nameSy that K falls within the triangle DLM and E outside the triangle DFM. These facts can be proved in the same way as the assumption in iii. 7. Let DE meet FM in K and LM in Z Then, as before, MZ is less than ML and therefore than MK, Therefore K lies further than Z from the foot of the perpendicular from M on DE. Similarly E lies further than Y from the foot of the same perpendicular. Heron deals with lines on opposite sides of the diameter through the external point in a manner similar to that adopted in his previous note. For the case where E, F sk the seeond points in which AE, AF meet the circle the method answers well enough. If AE is nearer the centre D than AF is, sqs. on DG, GE = sqs. on DH, HF and sqs, on DG^ GA = sqs, on DH, HA, whence, since it follows that and so that, by addition. DG < DM, GE>HF, AG>AIf, AE > AF.- But, if ^, Z be the points in which AE, A F first meet the circle, the method fails, and Heron is reduced to proving, in the first instance, the property usually deduced from 111. 36. He argues thus : AKD being an obtuse angle, sq, on AD = sum of sqs. on AK, KD and twice rect AK, KG. [». 11] ALD is also an obtuse angle, and it follows that sum of sqs. on AK, KD and twice rect, AK, KG is equal to sum of sqs. on AL, LD and twice rect. AL, LB. '" Therefore, the squares on KD, LD being eqjal, sq on AK ATid. twice rect AK, KG = sq. on AL and twice rect. AL, LH, or sq on AKm\6 rect. AK, ^£ = sq. on AL and rect. AL, LF, i.e. rect. AK, AE = reci. AL, AF. But, by the first part, AE > AF. Therefore AK<AL. in. 7, 8 deal with the lengths of the several lines drawn to the circum- ference of a circle {1) from a point within it, {2) from a point outside it; but a similar proposition is true of straight lines drawn from a point on the circumference itself: If any point be taken on the circumference of a circle^ then, ofalltht straight lines which can be drawn from it to the circumference, the greatest is that in which the centre is ; of any others that which is nearer to the straight line which passes through the centre is greater than one more remote ; and from the same point there can be drawn to the circumference two straight lines, and only t^vo, which are equal fo one another, one on each side of the greatest line. >ai' III. 8, 9] PROPOSITIONS 8, 9 ai The converses of in, 7, 8 and of the proposition just given are also true and can easily be proved by reducHo ad ahurdum. They could be employed to throw light on such questions as that of internal contact, and the relative position of the centres of circles so touching. This is clear when part of the converses is stated : thus (i) if from any point in the plane of a circle a number of straight lines be drawn to the circumference of the circle, and one of these is greater than any other, the centre of the circle must lie on that one, {1) if one of them is less than any other, then, (a) if the point is within the circle^ the centre is on the minimum straight line produced i^emi the point, (i) if the point is outside the circle, the centre is on the minimum straight line prioduced btyond the point in which if meets the drck. Proposition 9. !/ a point be taken within a circle, and more than two equal straight lines fall from the point on the circle, the point taken is the centre of the circle. Let ABC be a circle and D a point within it, and from D let more than two equal straight lines, namely DA, DB, DC, fall on the circle ABC ; I say that the point D is the centre of the circle ABC. For let AB, BC be joined and bisected at the points B, F, and let ED, FD be joined and drawn through to the points G, K, H, L. Then, since AE is equal to EB, and ED is common, the two sides AF, ED are equal to the two sides BE, ED ; and the base DA is equal to the base DB ; therefore the angle AED is equal to the angle BED. [1.8] Therefore each of the angles AED, BED is right ; [i. Def. lo] therefore GK cuts AB into two equal parts and at right angles. And since, if in a circle a straight line cut a straight line into two equal parts and at right angles, the centre of the circle is on the cutting straight line, [ni, i. Pot.) the centre of the circle is on GK. si BOOK III [hi. 9 For the same reason '■"" " ' ■ •'^ "^ * ■■ ' the centre of the circle ABC is also on HL. And the straight lines GK, HL have no other point common but the point D ; therefore the point D is the centre of the circle ABC. Therefore etc. Q. e. d. The result of this proposition is quoted by Aristotle, MettorolegUa in, 3, 373 a 13 — 16 (cf, note on i. 8). III. 9 is, as De Morgan remarks, a loguai equivalent of part of in. 7, where it is proved that every (w>«-centra.l point is not a point from which three equal straight lines can be drawn to the circle. Thus 111. 7 says that every nht-A is not-B, and in. 9 states the equivalent fact that every B \% A. Mr H. M. Taylor does in effect make a logical inference of the theorem that, If from a point three equal straight linei tan be drawn (0 a circle^ that point is the centre, by making it a corollary to his proposition which includes the part of in. 7 referred to. Euclid does not allow himself these logical inferences, as we shall have occasion to observe elsewhere also. Of the two proofs of this proposition given in earlier texts of Euclid, August and Heiberg regard that translated above as genuine, relegating the other, which Simson gave alone, to a place in an Appendix. Camerer remarks that the genuine proof should also have contemplated the case in which one or other of the straight lines AB, BC passes through D. This would however have been a departure from Euclid's manner of taking the most obscure case for proof and leaving others to the reader. The other proof, that selected by Simson, is as follows : " For let a point D be taken within the circle ABC, and from D let more than two equal straight lines, namely AD, DB, DC, fall on the circle ABC ; I say that the point D so taken is the centrt: of the circle ABC. For suppose it is not ; but, if possible, let it be £, and let D£ be joined and carried through to the points J^, G. Therefore fV is a diameter of the circle ABC. Since, then, on the diameter FG of the circle ABC a point has been taken which is not the centre of the circle, namely D, DG n greatest, and DC is greater than DB, and DB than DA, . But the latter are also equal : which is impossible Therefore E is not the centre of the circle. Similarly we can prove that neither is any other point except D; '•' ■ '-' therefore the point D is the centre of the circle ABC. "^ ' .... r ..I Q. E. D." On this Todhunter correctly points out that the point E might be supposed to fall within the angle ADC. It cannot then be shown that DC is greater than DB and DB than DA, but only that either i?C or DA is [ess than DB ; this however is sufficient for establishing the proposition. III. lo] PRur-OSlTIONS 9, lo t3 Proposition io, A circle does not cut a circle at more points than two. For, if possible, let the circle ABC cut the circle DBF at more points than two, namely B, C, /*, Ii \ let BH, BG be joined and bisected at the points K, L, and from K, L let KC, LM be drawn at right angles to BH, BG and carried through to the points A, E. Then, since in the circle ABC a straight line AC cuts a straight line BH into two equal parts and at right angles, the centre of the circle ABC is on AC. [in- i, For.] Again, since in the same circle ABC a straight line NO cuts a straight line BG into two equal parts and at right angles, the centre of the circle ABC is on NO. But it was also proved to be on AC, and the straight lines AC, NO meet at no point except at P ; therefore the point P is the centre of the circle ABC. Similarly we can prove that P is also the centre of the circle DEF\ therefore the two circles ABC, DBF which cut one another have the same centre P : which is impossible, [in- s] Therefore etc. q. e. d. I. The won) circle (niiXii)) ii here employed in the uousual (Case of the eireum/tremt {npt^ptM] of « drck. Cf. note on i. Der. ii,. There is nothing in the demonstration of this proposition which assumes that the circles cul one another ; it proves that two circles cannot mtet at mor^ than two points, whether they cut or meet without cutting, i.e. iouch one another, Hete again, of two demonstrations given in the earlier texts, Simson chos« the second, which Au(;u3t and Keilicrg relegate to an Appendix and which is as follows : " For again let the circle ABC cut the circle DEF at more points than two, namely B, G, H, F\ let the centre K of the circle ABC be taken, and let KB, KG^ KF be joined. BOOK III [hi, IO, II Since then a point K has been taken within the circle DEF, and from K more than two straight lines, namely KB, KF, KG, have fallen on the circle DEF, the point A' is the centre of the circle DEF. [in. 9] But K is also the centre of the circle ABC. Therefore two circles cutting one another have the same centre K : which is impossible, [111. 5] Therefore a circle does not cut a circle at more points than two. Q. E. D." This demonstration is claimed by Heron (see an-NairizI, ed, Curtie, pp. I JO — i). It is incomplete because it assumes that the point K which is taken as the centre of the circle ABC is within the circle DEF. It can however be completed by means of hi. 8 and the corresponding proposition with reference to a point on the circumference of a circle which was enunciated in the note on m. 8. For (i) if the point K is en the circumference of the circle DEF, we obtain a contradiction of the latter proposition which asserts that only two equal straight lines can be drawn from K to the circumference of the circle DEF; (i) if the point K is outside the circle DEF, we obtain a contradiction of the corresponding part of [ii. 8. Euclid's proof contains an unproved assumption, namely that the lines bisecting BG, BH at right angles will meet in a point P. For a discussion of this assumption see note on ir. 5. Proposition i i, If tivo circles touch one another internally, and their centres be taken, the straight line joining their centres, if it be also produced, will fall on the point of contact of the circles. For let the two circles ABC, ADE touch one another internally at the point A, and lei the centre F of the circle ABC, and the centre G of ADE, be taken ; I say that the straight line joined from G Xo F and produced will fall on A. For suppose it does not, but, if possible, let it fall hs, FGH, and let A F, AG he joined. Then, since J^G, 6"/^ are greater than FA, that is, than FH, let FG be subtracted from each ; therefore the remainder AG is greater than the remainder GH. HI. iij PROPOSITIONS lo, u »S But AG is equal to GD ; therefore GD is also greater than G//, the less than the greater : which is impossible. Therefore the straight line joined from F to G will not fall outside ; therefore it will fall at A on the point of contact. Therefore etc, Q. E. D. i. the straight line joining their centres, literally "the straight line joined to their 3. point of contact is here trtira^, and in the enunciation ur the next proposition Again August and Heiberg give in an Appendix the additional or alternative proof, which however shows little or no variation from the genuine proof and can therefore well be dispensed with. The genuine proof is beset with difficulties in consequence of what tt tacitly assumes in the figure, on the ground, probably, of its being obvious to the eye, Camerer has set out these difficulties in a most careful ^ote, the heads of which tnay be given as follows : He observes, first, that the straight line joining the centres, when produced, must necessarily (though this is not stated by Euclid) he produced in the dirtdion of the centre of the circle which touches the ether inltrnally. (For brevity, I shall call this circle the " inner circle," though I shall imply nothing by that term except thai it is the circle which touches the other on the inner side of the latter, and therefore that, in accordance with the definition of touching, points on it in the immediate neighbourhood of the point of contact are necessarily within the circle which it touches.) Camerer then proceeds by the following steps. T. The two circles, touching at the given point, cannot intersect at any f>oint. For, since points on the "inner" in the immediate neighbourhood of the point of contact are within the "outer" circle, the inner circle, if it intersects the other anywhere, must pass outside it and then return. This is only possible (o) if it passes out at one point and returns at another point, or {b) if it passes out and returns through one and the same point (a) is impossible because it would require two circles to have three common points ; {i) would require that the inner circle should have a node at the point where it passes outside the other, and this is proved to be impossible by drawing any radius cutting both loops. *. Since the circles cannot intersect, one must be entire^ within the other. 3, Therefore the outer circle must be greater than the inner, and the radius of the outer greater than that of the inner. 4. Now, if /■ be the centre of the greater and G of the inner circle, and if FG produced beyond G does not pass throt^h A, the given point of contact, then there are three possible hypotheses. ;. , (a) A may lie on GF produced beyond F. ' ■ ■ . 36 BOOK IIT [lit. II (#) A may lie outside the line FG altogether, in which case J^G produced beyond G must, in consequence of result 3 above, either (i) meet the circles in a point common to both, or {ii) meet the cirdes in two points, of which that which is on the inner circle is nearer to G than the other is. (a) is then proved to be impossible by means of the fact that the radius of the inner circle is less than the radius of the outer. {&) (ii) is Euclid's case ; and his proof holds equally of {#) (i), the hypothois, namely, that £> and Jf in the figure coincide- Thus all alternative hypotheses are successively shown to be impossible, and the proposition is completely established. I think, however, that this procedure may be somewhat shortened in the following manner. In order to make Euclid's proof absolutely conclusive we have only (i) to take care to produce /^G beyond G, the centre of the " inner " circle, and then (a) to prove that the point in which J^G so produced meets the *' inner " circle is nai further from G than is the point in which it meets the other circle. Euclid's proof is equally valid whether the first point is nearer to G than the second or the first point and the second coincide. If FG produced beyond G does not pass through A, there are two conceivable hypotheses : (a) A may lie on GF produced beyond F, or (i) A may be outside FG produced either way. In either case, if FG produced meets the " inner " circle in D and the other in H, and if GD is greater than GH, then the " inner " circle must cut the " outer " circle at some point between A and D, say X. But if two circles have a common point X lying on one side of the line of centres, they must have another conesponding point on the other side of the line of centres. This is clear from in. 7, 8 ; for the point is determined by drawing from F and G, on the opposite side to that where X is, straight lines FY, G Y making with FD angles equal to the angles DFX, DGX respectively. Hence the two circles will have at least three points common : which is impossible. Therefore GD cannot be greater than GH; accordingly GD must be either equal to, or less than, GH, and Euclid's proof is valid. The particular hypothesis in which FG is supposed to be in the same straight line with A but G is on the side of F^way from A is easily disposed of, and would in any case have been left to the reader by Euclid. For GD is either equal to or less than GH. Therefore GD is less than Fff, and therefore less than FA, But GD is equal to GA, and therefore greater than FA : which is impossible. in. II, i2j PROPOSITIONS n, la ay Subject to the same preliminary investigation as that required by Euclid's proof, the proposition can also be proved directly from iii. 7. For, by iii. 7, GH\% the shortest straight line that can be drawn from G to the circle with centre F; therefore GH\s less than GA, ' ' and therefore less than GD : which is absurd. This proposition is the crucial one as regards circles which touch internally; and, when it is once established, the relative position of the circles can be completely elucidated by means of it and the propositions which have preceded it. Thus, in the annexed f^ure, if ^ be the centre of the outer circle and G the centre of the inner, and if any radius FQ of the outer circle meet the two circles in Q, P respectively, it follows, from III. 7, in. 8, or the corresponding theorem with reference to a point on the circumference, that FA is the maximum straight line from .^to the circum- ference of the inner circle, FP is less than FA, and FP diminishes in length as FQ moves round from FA until FP reaches its minimum length FB. Hence the circles do not meet at any other point than A, and the distance PQ cut off between them on any radius FQ of the outer circle becomes greater and greater as FQ_ moves round from FA to FC and is a maximum when FQ coincides with FC, after which it diminishes again on the other side of FC. The same consideration gives the partial converse of in. 11 which forms the 6th lemma of Pappus to the first book of the Tactioms of Apollonius (Pappus, vn. p. 826). This is to the effect that, if h'&, AC art in ont straight lim, and on ont side of A, tht cirda described on AB, AC as diameters touch (internally at the point A). Pappus concludes this from the fact that the circles have a common tangent at A ; but the truth of it is clear from the fact that FP diminishes as FQ moves away from FA on either side ; whence the circles meet at A hut do not cut one another. Pappus' 5th lemma (vn. p. 824) is another partial converse, namely that, pven two circles touching internally at A, and a lint ABC drawn from A cutting both, then, if the centre of the outer circle lies on ABC, so does the centre -of the inner. Pappus himself proves this, by means of the common tangent to the circles at A, in two ways, (i) The tangent is at right angles to .^C and therefore to AB'. therefore the centre Qi the inner circle lies on AB. (2) By in. 32, the angles in the alternate segnients of both circles are right angles, so that ABC is a diameter of both. [Proposition 12. If two circles touch one another externally, the straight line joining their centres will pass through the point of contact. For let the two circles ABC, ADE touch one another S externally at the point A, and let the centre Foi ABC, and the centre G of ADE, be taken ; 38 - BOOK III [i[i. It I say that the straight line joined from F to G will pass through the point of contact at A. For suppose it does not, '" but, if possible, let it pass as FCDG, and let AF, AG be joined. Then, since the point F is the centre of the circle ABC, IS FA is equal to FC. Again, since the point G is the centre of the circle ADE, GA is equal to GD. But FA was also proved equal to FC ; » therefore FA, AG are equal to FC, GD, so that the whole FG is greater than FA, AG ; but it is also less [i. zo] : which is impossible, Therefore the straight line joined from F to G will not fail to pass through the point of contact at A ; n therefore it will pass through it. Therefore etc. >< • ' q, e. D.j ij. win not fall lo pang. The Greek has the doubk negaliye, o(ic dpe V-.'Wwli... adK iXedrirai, Literally ^' the straight line... will not ttef-^as,,.," Heron says on iii, 1 1 : " Euclid in proposition 1 1 has suppostKl the two circles to touch internally, made his proposition deal with this case and proved what was sought in it, Buf I will show how if is to be proved if the contact is external." He then gives substantially the proof and figure of ill, i». It seems clear that neither Heron nor an-Nairiit had ni. 1 2 in this place, Campanus and the Arabic edition of Naslraddin at-fQsI have nothing more of III. 12 than the following addition to 111. 11. "In the case of external contact the two lines ae and eb will be greater than ai, whence ad and cb will be greater than the whole ai, which is false." (The points a, b, c, d, e cor- respond respectively to G, P, C, D, A in the above figure.) It is most probable that Theon or some other editor added Heron's prt>of in his edition and made Prop. 12 out of it {an-Nairlit, ed, Curtze, pp. 121 — 2). An-NairM and Campanus, conformably with what has been said, number Prop. 13 of Hei berg's text Prop, i z, and so on through the Book. What was said in the note on the last proposition applies, mutatis mutandis, to this, Camerer proceeds in the same manner as before ; and we may use the same alternative argument in this case also. Euclid's proof is valid provided only that, if FG, joining the assumed centres, meets the circle with centre F in C and the other circle in D, C is not within the circle ADE and D is not within the circle ABC. {The proof is equally valid whether C, D coincide or the successive points are, as drawn in the figure, in the order F, C, D, G.) Now, if C is within the circle ADE III. I a] PROPOSITION ii 39 and D within the circle ABC, the circles must have cut between A and C and between A and D. Hence, as before, they must also have another corresponding point common on the other side of CO. That is, the circles must have three common points : which is impossible. Hence Euclid's proof is valid W F, A, G form a triangle, and the only hypothesis which has still to be disproved is the hypothesis which he would in any case have left to the reader, namely that A does not lie on FG but on FG produced in either direction. In this case, as before, either C, D must coincide or C is nearer /"than D is. Then the radius FC must be equal to FA : which is impossible, since FC cannot be greater than FD, and must therefore be less than FA. Given the same preliminaries, in, u can be proved by means of 111. 8, Again, when the proposition in. 12 is once proved, in, S helps us to prove at once that the circles He entirely outside each other and have no other common point than the point of contact. Among Pappus' lemmas to Aptollonius' Tactiones are the two partial converses of this proposition corresponding to those given in the last note. Lemma 4 (vii. p. 824) is to the effect that, tf AB, AC be in one straight tint, B and C bang on opposite sides 0/ A, the circles drawn on AB, AC as diameters touih externally at A. Lemma 3 (vii. p. 822) states that, 1/ two circles touch externally at A and BAC is drawn through h cutting both circles and containing the centre of one, BAC will also contain the centre of the other. The proofs, as before, use the common tangent at A. Mr H. M, Taylor gets over the difficulties involved by in. 11, 12 in a manner which is most ingenious but not Euclidean. He first proves that, j^rtco circles meet at a point not in the same straight line with their centres, the circles intersect at thai point ; this is very easily established by means of in. 7, 8 and the third similar theorem. Then he gives as a corollary the statement that, if two circles toueh, the point of contact is in the same straight line with their centres. It is not explained how this is inferred from the substantive proposition ; it seems, however, to be a logical inference simply. By the proposition, every A (circles meeting at a point not in the same straight line with the centre) is B (circles which intersect); therefore every not-^ is not' A, i.e. circles which do not intersect do not meet at a point not in the same straight line with the centres. Now non-intersecting circles may either meet {i.e. touch) or not meet. In the former case they must meet en the line of centres ; for, if they met at a point not in that line, they would intersect. But such a purely logical inference is foreign to Euclid's manner. As De Moi^an says, *' Euclid may have been ignorant of the identity of ' Every X is Y' and ' Every not- Y is noi-X,' for anything that appears in his writings ; he makes the one follow from the other by a new proof each time " (quoted in Keynes' Formal Legie, p. 8r), There is no difficulty in proving, by means of i. 20, Mr Taylor's next profMJsition that, if two circles meet at a point which lies in the same straight line as their centres and is between the centres, the circles touch at that point, and each circle lies without the ether. But the similar proof, by means of e. so, of the corresponding theorem for internal contact seems to be open to the same objection as Euclid's proof of in. 11 in that it assumes without proof that the circle which has its centre nearest to the point of meeting is the "inner" circle. Lastly, in order to prove that, if two circles hm'e a point of contact, they 30 BOOK III [ill. i> do not mat at any other point, Mr Taylor uses the qtiestionable corollary. Therefore in any case his alternative procedure doet not seem prefer&ble to Euclid's. The altcjrnative to Eucl. HI. ii — 13 which finds most favour in modem continental text-books (e.g. L^endre, Baltzer, Henrici and Treutlein, Veronese, Ingrami, Enriques and Amaldi) connects the number, position and nature of the coincidences between points on two circles with the relation in which the distance between their centres stands to the length of their radii. Enriqties and Amaldi, whose treatment of the different cases is typical, give the following propositions (Veronese gives them in the converse form). I . If the distance between the centres of two circles is greater than the stint of the radii, the two circles have no point common and are external to one another. Let O, ff be the centres of the circles (which we will call " the circles 0, O "), r, r their radii respectively. Since then OO >r-\- r', a fortiori OO ->r, and O is therefore exterior to the circle O. Next, the circumference of the circle intersects OG in a point A, and since 0O>r-¥r\ AO>r', and A is external to the circle O. But (7A is less than any straight line, as OB, drawn to the circum- ference of the circle O [in. 8] ; hence all points, as B, on the circumference of the circle are external to the circle ^- . . \ yo Lastly, if C be any point internal to the circle 0, the sum of (7C, f^C is ' greater than (/O, and a fortiori grtaX&r than r-\-r'. But OC is less than r: therefore OC is greater than t', or C is external to 0. Similarly we prove that any point on or within the circumference of the circle O is external to the circle 0- a. If the distance between the centres of two unequal circles is less than the difference of the radii, the two circumferences have no eontnton point and the lesser circle is entirety within the greater. Let 0, C be the centres of the two circles, r, r' their radii respectively Since Off <.r — r, a fortiori Off < r*, so that is internal to the circle O. If A, A' be the points in which the straight line 00 intersects respectively the circumferences of the circles 0, O, 00 is less than ffA'-OA, so that (7(3 + OA, or ffA, is less than OA', and therefore A is internal to the circle ff. But, of all the straight lines from O to the circumference of the circle O, OA passing through the centre O is the greatest [in, 7] ; whence all the points of the circumference of are internal to the circle O. A similar argument to the preceding will show that all points within the circle O are internal to the circle O. III. is] PROPOSITION 13 31 3. If the diitance btilvan the centra of two cirdti is equal to tht tuM of Iht radii, tht two drcumfirtnces have one point (ommon and one onfy, and that point is on the Um of an f res. Each circle is externa/ to the other. Let O, C be the centres, r, r the radii of the circles, so that OO \i equal \ar*-r'. Thus 00 is greater than r, so that O is external to the circle O, and the circum- ference of the circle O cuts OO in a point A. And, since OO is equal to /■ + r*, and OA to r, it follows that 0A is equal to r\ so that A belongs also to the circumference of the circle O. The proof that all other points on, and all points within, the circumference of the circle O are external to the cincle O follows the similar proof of prop. 1 above. And similarly all points (except A) on, and all points within, the circumference of the circle O are external to the circle O. The two circles, having one common point only, touch at that point, which lies, as shown, on the line of centres. And, since the circles are external to one another, they touch externally. 4. If the distance between the centres of two unequal circles is equal to tht difference between the radii, the two circumferences have one point and one only in common, and that point lies on the line of centres. The lesser circle it within tha other. The proof is that of prop. 2 above, mutatis mutandis. The circles here touch internally at the point on the line of centres. 5. If the distance between the centres of two circles is less than the sum, and greater than the di^erence, of the radii, the two circumferences hive two common points symmetrically situated with respect to the line of centres but not lying on that line. Let O, O \x the centres of the two circles, r, r their radii, *' being the greater, so that r'-r<Oa <r + *'. It follows that in any case 00 + /•> r', so that, if DM be taken on ffO produced equal to r (so that M is on the circumference of the circle 0), At is external to the circle ff.' We have to use the same Postulate as in Eucl. I, 1 that An arc of a circle which has one extremity within and the other without a given circle has one point common with the latter and only one ; from which it follows, if we consider two such arcs making a complete circumference, that, if a circum- ference of a circle passes through one point internal to, and one point external to a given circle, it cuts the latter circle in two points. We have then to prove that the circle O, besides having one point M of its circum- ference external to the circle ff, has one other point of its circumference (Z) internal to the latter circle. 3« BOOK in [ill. 11, 13 Three cases have to be distinguished according as 0^7 is greater than, equal to, or less than, the radius r of the lesser circle. (1) 00' > r, (See the preceding figure.) Measure OL along Off equal to r, so that Z lies on the circumference of the circle O. Then, since Off < r + ^, OL will be less than r, so that L is within the circle ff. {2) Off^r. In this case the circumference of the circle passes through ff, or L coincides with ff. (3) Off<^r. If we measure OL along Off equal to r, the point L will lie on the circumference of the circle O. Then OL^r- Off, so that O'L < r, and a fortiori ffL < r\ so that Z lies within the circle ff. Thus, in all three cases, since the circumference of O passes through one point {M) external to, and one point (L) internal to, the circle ff, the two circumferences intersect in two points A, B [Post. J And A, B cannot lie on the line of centres OO, since this straight tine intersects the circle O in L, M only, and of these points one is inside, the other outside, the circle O. Since AB\^a. common chord of both circles, the straight line bisecting it at right angles passes through both centres, i.e. is identical with Off ■ And again by means of 111. 7, 8 we prove that all points except A, B on the arc ALB lie within the circle ff, and all points except A^ B on the arc A MB outside that circle ; and so on. Proposition 13, A circle does not touch a circle at more points t/ian one, whether it touch it internally or externally. For, if possible, let the circle ABDC touch the circle EBFD, first internally, at more 5 points than one, namely D, B. Let the centre G of the circle ABDC, and the centre H of EBFD, be taken. Therefore the straight line 10 joined from G to /f will fall on B, D. [in. 11] Let it so fall, as BGHD. Then, since the point G is the centre of the circle A BCD, IS BG is equal to GD ; '^ m. 13] PROPOSITIONS ii, 13 3$ therefore BG is greater than ZfD ; » -- ' therefore B// is much greater than HD. Again, since the point // is the centre of the circle EBFD, ao B// is equal to //D ; but it was also proved much greater than it : which is impossible. Therefore a circle does not touch a circle internally at more points than one. .^^ *s I say further that neither does it so touch it externally. For, if possible, let the circle ACK touch the circle ABDC at more points than one, namely A, C, and let AC be joined. Then, since on the circumference of each of the circles JO ABDC, A CK two points A , C have been taken at random, the straight line joining the points will fall within each , circle ; [m. but it fell within the circle ABCD and outside ACK [ni. Def, 3] : which is absurd. js Therefore a circle does not touch a circle externally at more points than one. And it was proved that neither does it so touch it internally. Therefore etc. Q. e. d. 3, 7, [4, 37, 30. 33. ABDC Euclid writes ABCD (hew and in the next proposition), notwithstiuiding the order in which the points are placed in the iieure. tj, iT- does it so touch it. It is necessary to supply these words which the Gr«ek (Dri uliti licTin and fri oiSi itTij) leaves to be understood. The difficulties which have been felt in regard to the proofs of this proposition need not trouble us now, because they have already been disposed gf in the discussion of the more crucial propositions in. 1 1, 11. Euclid's proof of the first part of the proposition differs from Simson's ; and we will deal with Euclid's first. On this Cannerer remarks that it is assumed that the supposed second point of contact lies on the line of centres productd beyond the centre of tht "outer" circle, whereas all that is proved in III, 1 1 is that the line of centres produced beyond the centre of the " inner'" circle passes through a point of contact. But, by the same argument as that given on ni, 11, we show that the circles cannot have a point of contact, or even any common point, outside the line of centres, because, if there were such a point, there would be a corresponding common point on the other side of the line, and the circles would have three common points. Hence the only hypothesis left is that the second point of contact may be on the line of centres but in the direction of the centre of the "tfw^^r" circle; and Euclid's proof disposes of thb hypothesis. 34 BOOK in [ill. 13, 14 Heron (in an-Nairlzi, ed. Curtze, pp. m — 4), curiously enough, does not question Euclid's assumption chat the line of centres passes through both points of contact (if double contact is possible) ; but he devotes some space to proving that the centre of the "outer" circle must lie within the "inner" circle, a fact which he represents Euclid as asserting (" sicut dixit Euclides "), though there is no such assertion in our text. The proof of the fact is of course easy. If the line of centres passes through fe/A points of contact, and the centre of the "outer" circle lies either on or outside the "inner" circle, the line of centres must cut the "inner" circle in /hrei points in all: which is impossible, as Heron shows by the lemma, which he places here (and proves by i. 16), that a straight line cannot cut the circumfertnct of a circle in mere points than two. Simson's proof is as follows (there is no real need for giving two figures as he does). " If it be possible, let the circle EBF touch the circle ABC in more points than one, and first on the inside, in the points B, J?; join BD, and draw G/f bisecting B£> at right angles. Therefore, because the points B, D are in the circumference of each of the circles, the straight line BD falls within each of them : And their centres are in the straight line GH which bisects BD at right angles : Therefore GH passes through the Doint of ccMitact [ill. I ij ; but it does not pass through it, because the points B, D are without the straight line GH: which is absurd. Therefore one circle cannot touch another on the inside in more points than one." On this Camerer remarks that, unless ill. 11 be more completely elucidated than it is by Euclid's demonstration, which Slmson has, it is not sufficiently clear that, besides the point of contact in which GH meets the circles, they cannot have another point of contact either (1) on GH or (i) outside it. Here again the latter supposition (2) is rendered im possible because in that case there would be a third common point on the opposite side of GH ; and the former supf)osition,(i) is that which Euclid's proof destroys. Simson retains Euclid's proof of the second part of the proposition, though his own proof of the first part would apply to the second part also if a reference to iii. 12 were substituted for the reference to in. 11. Euclid might also have proved the second part by the same method as that which he employs for the first part. Proposition 14. In a circle equal straight lines are equally distant from the centre, and those which are equally distant from the centre are equal to one another. Let ABDC be a circle, and let ABt CD be equal straight lines in it ; I say that AB, CD are equally distant from the centre, For let the centre of the circle ABDC be taken [jh. \\ III. 14] PROPOSITIONS 13, 14 3^5 and let it be E\ from E let EF, EG be drawn perpendicular to AB, CD, and let AE, EC be joined. Then, since a straight line EF through the centre cuts a straight line AB not through the centre at right angles, it also bisects it. Therefore AF is equal to FB ; I therefore AB is double of AF, For the same reason I CD is also double of CG ; and AB is equal to CD ; therefore AF is also equal to CG. ' And, since AE is equal to EC, the square oxs. AE\^ also equal to the square on EC. But the squares on AF, EF^r^ equal to the square on AE, for the angle at F is right ; and the squares on EG, GC are equal to the square on EC, for the angle at G is right ; [r. 47] therefore the squares on AF, FE are equal to the squares on CG, GE, of which the square on AF is equal to the square on CG, for AFis equal to CG ; therefore the square on FE which remains is equal to the square on EG, therefore EF is equal to EG But in a circle straight lines are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal ; [ni. Def. 4] therefore AB, CD are equally distant from the centre. Next, let the straight hnes AB, CD be equally distant from the centre ; that is, let EF be equal to EG. I say that AB is also equal to CD. ~ ' ' For, with the same construction, we can prove, similarly, that AB is double of AF, and CD of CG. And, since AE is equal to CE, the square on AE is equal to the square on CE, But the squares on EF, FA are equal to the square on AE, and the squares on EG, GC equal to the square on CE. [i. 47] ^ BOOK in - -"^ [ill. 14, »5 Therefore the squares on £/^, FA are equal to the squares on £G, GC, i ■■ of which the square on EF is equal to the square on EG, for EF is equal to EG \ therefore the square on AF which remains is equal to the square on CG ; , , \ therefore ^^ is equal to CC " And y4^ is double of AF, and CD double of CG ; therefore AB is equal to CD. Therefore etc. Q. E. D. Heron (an-NairixI, pp. 125 — 7) has an elaborate addition to this proposition in which he proves, first by redtuiio ad aiiurdum, and then directly, that the centre of the circle falls between the two chords. Proposition 15. r,.. ... ,. • ..; 0/ straight lines in a circle the diameter is greatest, and of the rest the nearer to the centre is always greater than the more remote. Let ABCD be a circle, let AD be its diameter and E the centre ; and let BC be nearer to the , , diameter AD, and FG more remote ; I say that AD is greatest and BC greater than FG. For from the centre E let EH, EK be drawn perpendicular to BC, FG. Then, since BC is nearer to the centre and FG more remote, EK is greater than EH. [in. Def. 5] Let EL be made equal to EH, through L let LM be drawn at right angles to EK and carried through to N, and let ME, EN, FE, EG be joined. Then, since EH is equal to EL, BC is also equd to MN. [m. 14] Again, since AE is equal to EM, and ED to EN, AD is equal to ME, EN. III. IS, 16} PROPOSITIONS 14—16 37 But ME, EN are greater than MN, [1. ao] and MN is equal to BC\ ' ' therefore AD is greater than BC. And, since the two sides ME, EN are equal to the two sides FE, EG, and the angle MEN greater than the angle FEG, therefore the base MN is greater than the base FG, [i. 14] But MN was proved equal to BC. Therefore the diameter AD is greatest and BC greater than FG. Theretore etc. g. e. d, 1. Of straight lines. The Greek leaves these words to be understood. 5. Nearer to the diameter AD, As BC, FG are not In general parallel to AD, Euclid should have said ^' nearer to the centre." It will be observed that Euclid's proof differs from that given in our text- books (which is Simson's) in that Euclid introduces another line MN, which is drawn so as to be equal to BC but at right angles to EK and therefore parallel to FG. Sim son dispenses with MN^vA tases his proof on a similar proof by Theodosius {Spkatrica i, 6). He proves that the sum of the squares on EH, HB is equal to the sum of the squares on EK, KF\ whence he infers that, since the square on EH'\% less than the square on EK, the square on BH is greater than the square on FK. It may be that Euclid would have regarded this as too complicated an inference to make without explanation or without an increase in the number of his axioms. But, on the other hand, Euclid himself assumes that the angle subtended at the centre by MN is greater than the angle subtended by FG, or, in other words, that M, N both fall outside the triangle FEG. This is a similar assumption to that made in lit, 7, 8, as already noticed; and its truth is obvious because EM, EN, being r<idii of the circle, are greater than the distances from E to the points in which MN cuts EF, EG, and therefore the latter points are nearer than M, N&it to Z, the foot of the perpendicular from E to MN. Simson adds the converse of the proposition, proving it in the same way as he proves the proposition itself. Proposition i6. .r ••'•:, TAe slraighl line dragon ai right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed ; further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilineal angle. Let ABC be a circle about D as centre and AB as diameter ; 3S BOOK in fill. i6 I say that the straight line drawn from A at right angles to AB from its extremity will fall -^ , outside the circle. For suppose it does not, but, if possible, let it fall within as CA, and let DC be joined. Since DA is equal to DC, the angle DAC is also equal to the angle A CD, [i. s] But the angle DAC is right ; therefore the angle ACD is also right : thus, in the triangle ACD, the two angles DAC, ACD are equal to two right angles : which is impossible. [i. 1 7] Therefore the straight line drawn from the point A at right angles to BA will not fall within the circle. Similarly we can prove that neither will it fall on the circumference ; therefore it will fall outside. Let it fall as AE ; I say next that into the space between the straight line AE and the circumference CHA another straight line cannot be interposed. For, if possible, let another straight line be so interposed, as EA, and let DG be drawn from the point D perpendicular to EA. -, Then, since the anrie A GD is right, ' and the angle DA G is less than a right angle, AD is greater than DG. [i. 19] But DA is equal to DH ; therefore DH is greater than DG, the less than the greater : which is impossible. Therefore another straight line cannot be interposed into the space between the straight line and the circumference, I say further that the angle of the semicircle contained by the straight line BA and the circumference CHA is greater than any acute rectilineal angle, and the remaining angle contained by the circumference CHA and the straight line AE is less than any acute rectilineal angle. For, if there is any rectilineal angle greater than the angle contained by the straight line BA and the circumference III. 16] PROPOSITION i6 39 CHA, and any rectilineal angle less than the angle contained by the circumference CHA and the straight line AE, then into the space between the circumference and the straight line AE a straight line will be interposed such as will make an angle contained by straight lines which is greater than the angle contained by the straight line BA and the circumference CHA, and another angle contained by straight lines which is less than the angle contained by the circumference CHA and the straight line AE. But such a straight line cannot be interposed ; therefore there will not be any acute angle contained by straight lines which is greater than the angle contained by the straight line BA and the circumference CHA, nor yet any acute angle contained by straight lines which is less than the angle contained by the circumference CHA and the straight line AE. — PoKiSM. ' From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its extremity touches tne circle. ,. cannot be Interposed, Ut«ir>lly " will not fall in between" (od nptikwaCrat). This proposition is historically interesting because of the controversies to which the last part of it gave rise from the 13th to the 17th centuries. History was here repeating itself, for it is certain that, in ancient Greece, both before and after Euclid's time, there had been a great deal of the same sort of contention about the nature of the " angle of a semicircle " and the "remaining angle" between the circumference of the semicircle and the tangent at its extremity. As we have seen (note on i. Def. 8), the latter angle had a recognised name, iMparoitSij? yuivii, hern-tike or eomitu!ar angle ; though this term does not appear in Euclid, it is often used by Proctus, evidently as a term well understood. While it is from Proclus that we get the best idea of the ancient controversies on this subject, we may, I thinl^ infer their prevalence in Euclid's time from this solitary appearance of the two " angles " in the Elements. Along with the definition of the angle 0/ a segment, it seemi. to show that, although these angles are only mentioned to be dropped again immediately, and are of no use in elementary geometry, or even at all, Euclid thought that an allusion to them would be expected of him ; it is as if he merely meant to guard himself against appearing to ignore a subject which the geometiers of his time regarded with interest. If this conjecture b right, the mention of these angles would correspond to the insertion of definitions of which he makes no use, e.g. those of a rhombus and a rhomboid. Proclus has no hesitation in speaking of the " angle of a semicircle " and the "hom.like angle" as true angles. I'hus he says that "angles are contained by ■i. straight line and a circumference in two ways ; for they are either contained by a straight line and a cunve.Y circumference, like, that of the setni- 40 BOOK III [ni. 1 6 circle, or by a straight line and a concave circumference, like the mparodSift " (p. 127, II — 14). "There are mixed lines, as spirals, and angles, as the angle of a semicircle and the ntpaTOfiSij! " (p. 104, 16—18). The difficulty which the ancients felt arose from the very fact which Euclid embodies in this proposition. Since an angle can be divided by a line, it would seem to be a magnitude; "but if it is a magnitude, and all homogeneous magnitudes which are finite have a ratio to one another, then alt homogeneous angles, or rather all those on surfaces, will have a ratio to one another, so that the cornitular will also have a ratio to the rectilineal. But tilings which have a latio to one another can, if multiplied, exceed one another. Therefore the cornteular angle will also sometime exceed the rectilineal ; which ts impossible, for it is proved that the former is less than any rectilineal angle'' (Proclus, p. lai, 24 — I2Z, 6). The nature of contact between straight lines and circles was also involved in the question, and that this was the subject of controversy before Euclid's time is clear from the title of a work attributed to Democritus (13. 420 — 400 B.C.) irtpi £((u^ap^i^9 yvu^oi/ot ^ irfpl ^axaxjOK icuicAtnf fmx trtpaiptjif On a differenu in a gnomon or on eontaei of a drcU and a sphert. There is, however, another reading of the first words of this title as given by Diogenes I^aertius (ex. 47), namely iripl Sta^p^i yfilfLiTt. On a difference of opinion, etc. May it not be that neither reading is correct, but that the words should be TTtpt Sto^p^i ytuci't/t t) jrtpt ^aiicTMx mJicXou not <r^<upt)!. On a difference in an angle or on contact with a circle and a sphere} There would, of course, hardly be any "angle" in conne>tion with the sphere; but I do not think that this constitutes any difficulty, because the sphere might easily be tacked on as a kindred subject to tiie circle. A curiously similar collocation of words appears in a passage of Proclus, though this may be an accident. He sa^s {p. 5*^* 4) "^^ ^i y*avi^v Bia^opOr^ ktyofJ4v nai aiffi^ftf auruit' ... and then, in the next hne but one, tt^ &i rav a^s rtay kukXiov ^ rmv cwfuiiv, " In what sense do we speak of differences of angles and of increases of than . . . and in what sense of the contaets (or meetings) of circles or of straight lines ? " I cannot help thinking that this subject of comicular angles would have had a fascination for Democritus as being akin to the question of infinitesimals, and very much of the same character as the other question which Plutarch {On Common Notions, xxxix. 3) says that he raised, namely that of the relation between the base of a cone and a section of it by a plane parallel to the base and apparenrty, to judge by the context, infinitely near to it : " if a cone were cut by a plane parallel to its base, what must we think of the surfaces of the sections, that they are equal or unequal? For, if they are unequal, they will make the cone irregular, as having many indentations like steps, and unevennesses ; but, if they are equal, the sections will be equal, and the cone will appear to have the property of the cylinder, as being made up of equal and not unequal circles, which is the height of absurdity." The contributions by Democritus to such investigations are further attested by a passage in the Method of Archimedes discovered by Heiberg In 1906 {Archimedes, ed. Heiberg, Vol. ii. 191 3, p. 430; T. L. Heath, Tl'e Method of Archimedes, 1912, p. 13), which says that, though Eudoxus was the first to discover the scientific proof of the propositions (attributed to him) that the cone and the pyramid are one-third of the cylinder and prism respectively which have the same base and equal height, they were first stated, without proof, by Democritus. A full history of the later controversies about the cornicuiar " angle " cannot be given here ; more on the subject will be found in Camerer's Euclid (Excursus iv. on 11 1. 16) or in Cantor's Gtschichte der Maihematik. III. id} PROPOSITION i6 41 Vol. 11. {see Contingenzwmkti in the index). But the following short note about the attitude of certain well-known mathematicians to the question will perhaps not be out of place, Johannes Campanus, who edited Euclid in the 13th century, inferred from [ti. 16 that there was a flaw in the principle that the iransitien from the less to the greater, or vice vers A, fakes place through all intermediate quantities and therefore through the egxial. If a diameter of a circle, he says, be moved about its extremity until it takes the position of the tangent to that circle, then, as lon^ as it cuts the circle, it makes an acute angle less than the " angle of a semicircle " ; but the moment it ceases to cut, it niakes a right angle greater than the same " angle of a semicircle." The rectilineal angle is never, during the transition, egual to the " angle of a semi- circle." There is therefore an apparent inconsistency with x. 1, and Campanus could only observe (as he does on that proposition), in explanation of the paradox, that " these are not angles in the same sense (univoce), for the curved and the straight are not things of the same kind without qualification (simpliciter)." The argument assumes, of course, that the right angle is greater than the "angle of a semicircle." Very similar is the statement of the paradox by Cardano (1501 — 1576), who observed that a quantity may eoutifiually increase without limit, and another diminish •without limit ; and yet the firsts however iTiereased, may be lest than the second, however diminished. The first quantity is of course the angle of contact, as he calls it, which may be " increased " indefinitely by drawing smaller and smaller circles touching the same straight line at the same point, but will always be less than any acute rectilineal angle however small. We next come to the French geometer, Peletier (Peletarius), who edited the Elements in r 557, and whose views on this subject seem to mark a great advance. Peletier's opinions and arguments are most easily accessible in the account of them given by Clavius (Christoph Klau[?], 1537 — 1612) in the 1607 edition of his Euclid. The violence of the controversy between the two will be understood from the fact that the arguments and counter-ai^uments (which sometimes run into other matters than the particular question at issue) cover, in that book, xt pages of small print. Peletier held that the " angle of contact " was not an angle at all, that the "contact of two circles," i.e. the "angle" between the circumferences of two circles touching one another internally or externally, is not a quantity, and that the " contact of a straight line with a circle " is not a quantity either; that angles contained by a diameter and a circumference whether inside or outside the circle are right angles and equal to rectilineal right angles, and that angles contained by a diameter and the circumference in all circles are equal The proof which Peletier gave of the latter pro- position in a letter to Cardano is sufficiently ingenious. If a greater and a less semicircle be placed with their diameters terminating at a common point and lying in a straight line, then (i) the angle ^the larger obviously cannot be less than the angle of the smaller. Neither (*) can the former be greater than the latter ; for, if it were, we could obtain another angle of a semicircle greater still by drawing a still larger semicircle, and so on, until we should ultimately have an angle of a. semicircle greater than a right angle ; which is imp)OSsible. Hence the angles ^semicircles must all be equal, and the dif- ferences between them nothing. Having satisfied himself that all angles of contact are JftfAangles, no^qu an titles, and therefore nothings, Peletier holds the difficulty about x. i to be at an end. He adds the interesting remark that the essence of an angle is in cutting, not contact, and that a tangent is not inclined to the circle at the point of contact but is, as it were, immersed in it at that point, just as much as if the circle did not diverge from it on either side. 4» BOOK III [ill. 16 The reply of Claviua need not detain us. He argues,' evidently appealing to the eye, that the angle of contact qan be divided by the arc of a circle greater than the given one, that the angles of two semicircles of different sizes cannot be equal, since they do not coincide if they are applied to one another, that there is nothing to prevent angia of coniact from being quantities, it being only necessary, in view of x. i, to admit that they are not of the same kind as rectilineal angles ; lastly that, if the angle of contact had been a nothing, Euclid would not have given himself so much trouble to prove that it is less than any acute angle. {The word is dtsudasset, which is certainly an exaggeration as applied to what is little more than an obiter dictum in in. id.) Vieta (1540 — 1603) ranged himself on the side of Peletter, maintaining that the angle of contact is no angle ; only he uses a new method of proof. The circle, he says, may be regarded as a plane figure with an infinite number of sides and angles ; but a straight line touching a straight line, however short it may be, will coincide with that straight line and wilt not make an angle. Never before, says Cantor (ii,, p. 540), had it been so plainly dccKired what exactly was to be understood by contact, Gahleo Galilei (1564 — 164*) seems to have held the same view as Vieta and to have supported it by a very similar argument derived from the com- parison of the circle and an inscribed polygon with an infinite number of sides. The last writer on the question who must be mentioned is John Wall is (1616— 1703}. He published in 1656 a paper entitled De angttlo contactus et semicireuli traetatus in which he also maintained that the so-cailed angle was not a true angle, and was not a quantity, Vincent Leotaud (1555—1672) took up the cudgels for Clavius in his Cyclomathia which appeared in 1663, This brought a reply from Wallis in a letter to Leotaud dated 17 February, 1667, but not apparently published till it appeared in A defense of the treatise of the angle of contact which, with a separate title-page, and date 1G84, was included in the English edition of his Algebra dated 1685, The essence of Wallis' position may be put as follows. According to Euclid's definition, a plane angle is an inclination of two lines; therefore two lines forming an angle must incline to one another, and, if two lines meet without being inclined to one another at the point of meeting (which is the case when a circumference is touched by a straight line), the lines do not form an angle. The " angle of contact " is therefore no angle, because at the point of contact the straight line is not inclined to the circle but lies on it (1kA.lv<j9, or is coincident with it. Again, as a point is not a line but a heginning of a line, and a line is not a surface but a beginning oi a surface, so an angle is not the distance between two lines, but their initial tendency towards separation : Angulus {seu gradus divaricatianis) Distantia nen est sed Incef/thius distantiae. How far lines, which at their point of meeting do not fomt an angle, separate from one another as they pass on depends on the degree of curvature (gradus curvitatis), and it is the latter which has to be compared in the case of two lines so meeting. The arc of a smaller circle is more curved as having as much curvature in a lesser length, and is therefore curved in a greater degree. Thus what Clavius called angulus contactus becomes with Wallis gradus curvitatis, the use of which expression shows that curvature and curvature can be compared according to one and the same standard. A straight line has the least possible curvature ; but of the "angle" made by it with a curve which it touches we cannot say that it is greater or less than the " angle " which a second curve touching the same straight line at the same point makes with the first curve ; for in both ca.<«s there is no true angle at all (cf. Cantor m,, p. 24). lit. 16, 17] PROPOSITIONS 16, 17 43 The words usually given as a part of the corollary "and that a straight line touches a circle at one point only, since in fact the straight line meeting it in two points was proved to fall within it " are omitted by Hetberg as being an undoubted addition of Theon's. It was Simson who added the further remark that "it is evident that there can be but one straight line which touches the circle at the same point" Proposition 17. From a given point to draw a straight line touching a given circle. Let A be the given point, and BCD the given circle ; thus it is required to draw from the point A a straight line touching the circle BCD. For let the centre E of the circle be taken ; [m, i] let AE be joined, and with centre E and distance EA let the circle AEG be described ; from D Jet DF be drawn at right angles to EA, and let £F, AB h& joined ; I say that AB has been drawn from the point A touching the circle BCD. For, since E is the centre of the circles BCD, AFG, EA is equal to EF, and ED to EB ; therefore the two sides AE, EB are equal to the two sides FE, ED : and they contain a common angle, the angle at E ; therefore the base DF is equal to the base AB, .1 and the triangle DEE is equal to the triangle BE A, and the remaining angles to the remaining angles ; [i. 4] therefore the angle EDF is equal to the angle EBA. But the angle EDF is right ; therefore the angle EBA is also right. Now EB is a radius ; , and the straight line drawn at right angles to the diameter of a circle, from its extremity, touches the circle ; [in. 16, Por,] therefore AB touches the circle BCD. Therefore from the given point A the straight line AB has been drawn touching the circle BCD. m / BOOK HI [ill. 17, 18 The construction shows, of course, that two straight lines can be drawn from a givttn external point to touch a ^ven circle ; and it is equally obvious that these two straight lines are equal in length and equally inclined to the abaight line joining the exiemai point to the centre of the given circle. These facts are given by Heron {an-Nairlzl, p. 130). It is true that Euclid leaves out the case where the given point lies oit the circumference of the circle, doubtless because the construction is so directly indicated by iii. 16, For. as to be scarcely worth a separate statement. An easier solution is of course possible as soon as we know (ici. 31) that the angle in a semicircle is a right angle ; for we have only to describe a circle on AE as diameter, and this circle cuts the given circle in the two points of contact Proposition 18. // a straight tine touch a circle, and a straight line be jained from the centre to the point of cop fact, the straight line so joined wilt be perpendicular to the tangent. For let a straight line D£ touch the circle ABC at the point C let the centre F of the circle ABC be taken, and let FC be joined from F\o C; I say that FC is perpendicular to DE. For, if not, let FG be drawn from F perpendicular to DE. Then, since the angle FGC is right, the angle FCG is acute ; [i. 1 7] and the greater angle is subtended by the greater side ; [1. 19] therefore FC is greater than FG. But FC is equal to FB ; therefore FB is also greater than FG, the less than the greater: which is impossible. Therefore FG is not perpendicular to DE. Similarly we can prove that neither is any other straight line except FC ; therefore FC is perpendicular to DE. Therefore etc. Q. E, D. til, i8, 19] PROPOSITIONS 17—19 4S 3. the tangnit, 4 ifuwreiUr^. Just as 111. 3 contains two partial converses of the ForUm to lit. i, so the present proposition and the next give Jwo partial converses of the corollary to iii, 16, We may show their relation thus: suppose three things, ( r) a tangent at a jwint of a circle, (2) a straight line drawn from the centre to the point of contact, (t) right angles made at the point of contact [with (i) or {1) as the case may bej. Then the corollary to in. 16 asserts that (t) and (3) together give (i), iii. 18 that (t) and (i) give (3), and iii. rg that (t) and (3) give (1), i.e. that the straight line drawn from the point of contact at right angles to the tangent passes through the centre. Proposition 19. If a straight line iottch a circle, and from the point of contact a straight line be drawn at right angles to the tangent, the centre of the circle will lie on the straight line so dratvn. For let a straight line DE touch the circle ABC at the point C, and from C let CA be drawn at right angles to DE ; I say that the centre of the circle is on A C. For suppose it is not, but, if possible, let F be the centre, and let CF be joined. Since a straight line DE touches the circle ABC, and FC has been joined from the centre to the point of contact, FC is perpendicular to DE ; [111, 18] therefore the angle FCE is right But the angle ACE is also right ; therefore the angle FCE is equal to the angle ACE, the less to the greater : which is impossible. < Therefore F is not the centre of the circle ABC. Similarly we can prove that neither is any other point except a point on AC. Therefore etc. Q. E. D. We may abo regard iii, 19 as a partial converse of in. 18. Thus suppose (t) a straight line through the centre, (s) a straight line through the point of contact, and suppose (3) to mean perpendicular to the tangent ; then iii. tS asserts that (i) and (2) combined produce (3}, and 111. 19 that (1} and (3) 4& / »• BOOK Iir ^.1'' [111.19,30 produce (i); while again we may enundate a second partial converse of iii, 18, corresponding to the statement that (i) and (3) produce (2), to the effect that a straight line drawn through the centre perpendicular to the tangent passes through the point of contact. We may add at this poin^ or even after the Porism to ill, 16, the theorem that ttuo circles which touch om another internally or externally have a common tangent at their point of cotttaei. For the line joining their centres, produced if necessary, passes through their point of contact, and a straight line diawn through that point at right angles to the line of centres is a tangent to both circles. Proposition 20. In a circle the angle at ihe centre is double of the angle at the circumference, when the angles have the same circum- ference as base. Let ABC be a circle, let the angle BEC be an angle sat its centre, and the angle BAC an angle at the circumference, and let them have the same circumference BC as base ; I say that the angle BEC is double of 10 the angle BAC. For let AE be joined and drawn through to F. Then, since EA is equal to EB, the angle EAB is also equal to the IS angle EBA ; [1. 5] therefore the angles EAB, EBA are double of the angle EAB. But the angle BEF is equal to the angles EAB, EBA ; ['■ 3-1 therefore the angle BEF is also double of the angle taEAB. ' For the same reason the angle FEC is also double of the angle EAC. Therefore the whole angle BEC is double of the whole angle BAC. as Again let another straight line be inflected, and let there be another angle BDC\ let DE be joined and produced to G, 111. »o] PROPOSITIONS 19, 10 47 Similarly then we can prove that the angle GEC is double of the angle EDC, » of which the angle GEB is double of the angle EDB ; therefore the angle BEC which remains is double of the angle BDC. Therefore etc. Q. e. d. 15. let another straight line be inflected, atM.tin ik ir(Ui> (without (Wcia). The verb jfXdw (to brtak off) was the regular technical term for drawln^r from a point a (broken) straight line which hfst tneeis another straight line or curve and is then htnt lnuk ham it to anmher point, or (in other words) for drawing .straight lines from two points meeting at a point on a curve or another straight line. Ki>t\Aff9v^t Li one of the geometrical terms ^he definition of which must according to Aristotle be assumed [AtuiL Rat, t. \o, 76 b 9). The early editors, Tartaglia, Commandinus, Peletarius, Clavius and others, gave the extension of (his proposition to the case where the segment is less than a semicircle, and where accordingly the " angle " corresponding to Euclid's " atigle at the centre " is greater than two right angles. The convenience of the extension is obvious, and the proof of it is the same as the first part of Euclid's proof. By means of the extension in. 2 1 is demonstrated without making two cases; Jti. zz will follow immediately from the fact that the sum of the " angles at the centre " for two segments making up a whole circle is equal to four right angles; also 111. 31 follows immediately from the extended proposition. But all the editors referred Xq were forestalled in this matter by Heron, as we now learn from the commentary of an-Naitizi (ed. Curtxe, p. 131 sqq.). Heron gives the extension of Euclid's pro[>osition which, he says, it had been left for him to make, but which is necessary in order that the caviller may not be able to say that the next proposition (about the equality of the angles in any segment) is not established generally, i.e. in the case of a segment less than a semicircle as well as in the case of a segment greater than a semicircle, inasmuch as lit. no, as given 'ay Euclid, only enables us to prove it in the latter case. Heron's enunciation is imt>ortaiit as showing how he describes what we should now call an " angle " greater than two right angles. (The language of Gherard's translation is, in other respects, a little obscure ; but tht: meaning is made clear by what follows.) "The angle," Heron says, "which is at the centre of any circle is double of the angle which is at the circumference of it vrhen one arc is tk( base of bolk angles; and the remaining angles whieh are at I he centre, and fill up the four right angles, are double of the angle at the circumference of the ate which is subtended by the [original] angle which is at the ceiitre," Thus the " angle greater than two right angles " is for Heron the sum of certain "angles" in the Euclidean sense of angles less than two right angles. The particular method of splitting up which Heron adopts will be seen from his proof, which is in substance as follows. r Let CDB be an angle at the centre, CAB that at the circumference. ' Produce SD, CDto F,G; take any point jE on BC^ and join BE, EC, ED. Then any angle in the segment BAC is half of the angle SDC; and tht turn of the angles BDG, GDF, FDC is double of any angle in the segment BEC. 48 -I BOOK in [til. 10 I'ttof. Since CZ7 is equal, to £Z7, I't./ ■• ,tuiM v i - the angles DCE, DEC art equal. Therefore the exterior angle GDE is equal to twice the angle DEC. Similarly the exterior angle EDE is equal to twice the angle DEB. By addition, the angles GDE, EDE are double of the angle BEC. But the angle BDC is equal to the angle EDG, therefore the sum of the anglts BDG, GDF, FDC is doubU of the angle BEC. And Euclid has proved the first part of the proposition, namely that the angle BDC is double of the angle BAC. Now, says Heron, BAC is any angle in the segment BAC, and therefore any angle in the segment BAC is half of the angle BDC. Therefore all the angles in the segment BAC are equal. Again, BEC is any angle in the segment BEC and is equal to half the sum of the angles BDG, GDF, FDC. Therefore all the angles in (he segment BEC are equal Hence in. 2 1 is proved generally. Lastly, says Heron, since the sum of the angles BUG, GDF, FDC is double of the angle BEC, and the angle BDC is double of the angle BAC, therefore, by addition, the sum of four right angles is double of tKe sum of the angles .^.^C, BEC. Hence the angles BAC, BEC are together equal to two right angles, and III. 12 is proved. The above notes of Heron show conclusively, if proof were wanted, that Euclid had no idea of in. zo applying in terms (either as a matter of enunciation or proof) to the case where the angle at the circumference, or the angle in the segmenl^ is oituse. He would not have recognised the " angle " greater than two right angles or the so-called "straight angle" as being an angle at all. This is indeed clear from his definition of an angle as the ittciinatien ic.r.i,, and from the language used by other later Greek mathe- maticians where there would be an opportunity for introducing the extension. Thus Proclus' notion of a "four-sided triangle" (cf the note above on the definition of a triangle) shows that he did not count a re-entrant angle as an angle, and Zenodorus' application to the same figure of the word "hollow- angled " shows that in that case it was the exterior angle only which he would have called an angle. Further it would have been inconvenient to have introduced at the beginning of the Elements an "angle" equal to or greater than two right angles, because other definitions, e;g. that of a right angle, would have needed a qualification. If an "angle" might be equal to two right angles, one straight line in a straight line with another would have satisfied Euclid's definition of a right angle. This is noticed by Dodgson (p. 160), but it is pmctically brought out Dy Proclus on i, 13. "For he did not merely say that ' any straight line standing on a straight line either makes two right angles or angles equal to two right angles ' but ' if it make angles.' III. 20, »i) PROPOSITIONS 30, Ji 49 If it stand on the straight line at it$ extremity and make one angle, is it possible for this to be equal to two right angles ? It is of course impossible ; jbr every rectUineai angle is iess than two right angles, as every solid angle is less than four right angles (p. 291, 13 — 20)." [It is (rue that it has been generally held that the meaning of " angle " is tacitly extended in vi. 33, but there is no real ground for this view. See the note on the proposition.! It will be observed that, following his usual habit, Euclid omits the demonstration of the case which some editors, e.g. Clavius, have thought it necessary to give separately, the case namely where one of the lines forming the angle in the s^ment passes through the centre. Euclid's proof gives so obviously the means of proving this that it is properly left out. Tod hunter observes, what Clavius had also remarked, that there are two assumptions in the proof of 111. zo, namely that, if A is double of B and C double of D, then the sum, or difference, of A and C is equal to double the sum, or difference, of B and D respectively, the assumptions being particular cases of v. i and v. 5. But of course it is easy to satisfy ourselves of the correctness of the assumption without any recourse to Book v. ' ' ' Proposition 21. In a circle the angles in ike same segment are equal to one another. Let A BCD be a circle, and let the angles BAD, BED be angles in the same segment BAED ; I say that the angles BAD, BED are equal to one another. For let the centre of the circle ABCD be taken, and let it be .f ; let BE, ED be joined. Now, since the angle BED is at the centre, and the angle BAD at the circum- ference, and they have the same circumference BCD as base, therefore the angle BED is double of the angle BAD. [m. 20] For the same reason the angle BFD is also double of the angle BED ; therefore the angle BAD is equal to the angle BED. / Therefore etc, , Q. E. D. Under the restriction that the " angle at the centre " used in iii, *o must be less than two right angles, Euclid's proof of this proposition only applies to the case of a segment greater than a semicircle, and the case of a segment equal to or less than a semicircle has to be considered separately. The simplest proof, of many, seems to be that of Simson. s* BOOK in [hi. 21 " But, if the segment BARD be not greater than a semicircle, let SAD, BED be angles in it: these also are equal to one another. Draw AJ^lQ the centre, and produce it to C, and join CE. Therefore the segment BADC is greater than a semicircle, and the angles in it SAC, EEC are equal, by the first case. For the same reason, because CBED is greater than a semicircle, the angl^ CAD, CED are equal. Therefore the whole angle BAD is equal to the whole angle BED." We can prove, by means of redudio ad absurdiim, the important converse of this proposition, namely that, if there be any hoo Iriangies on the same base and on the same side of it, and with equal vertical angles, the circle passing through the extremities of the base and the vertex of one triangle will pass through the vertex of the other triangle also. That a circle can be thus described about a triangle is clear from Euclid's construction in itt, 9, which shows how to draw a circle passing through any three points, though it is in tv. 5 only that we have the problem stated. Now, suppose a circle BAC drawn through the angular points of a triangle BAC, and let BDC be another triangle with the same base BC and on the same side of it, and having its vertical angle D equal to the angle A. Then shal! the circle pass through D. For, if it does not, it must pass through some point E on BD or on BD produced. If then EC be joined, the angle BEC is equal to the angle BAC, by in. %\, and therefore equal to the angle BDC. Therefore an exterior angle of a triangle is equal to the interior and opposite angle; which is impossible, by 1. 16. Therefore D lies on the circle BA C. Similarly for any other triangle on the base BC and with vertical angle equal to A. Thus, if any number of triangles be constructed on the same base and on the same side of it, with equal vertical angles, the vertices will all lie on the circumfererue of a segmetit of a circle. A useful theorem derivable from ill. 21 is given by Serenus (ZV sectUme coni. Props. 5?, 53). If ADB be any segment of a circle, and C l>e such a point on the circumference that AC \i equal to CB, and if there be described with C as centre and radius CA or CB the circle AI/B, then, ADB being any other angle in the segment ACB, and BD being produced to meet the outer segment in E, the sum of AD, DB is equal to BE. If BC be produced to meet the outer segment in F, and FA be joined, CA, CB, CEaie by hypothesis equal. Therefore the angle EAC is equal to the angle AEC. Also, by Itt. It, the angles ACB, ADB are equal ; III. ai, 12] PROPOSITIONS 11, a* 51 therefore their supplements, the angles jiC/^, AJ}£, are equal Further, by m. ai, the angles AEB, AFB are equal. Hence in the triangles ACF, ADE two angles are respectively equal ; therefore the third angles EAD, FAC are equal. But the angle FAC is equal to the angle AFC, and therefore equal to the angle AED. Therefore the angles AED, EAD are equal, or the triangle DEA is isosceles, and AD is equal to DE. Adding BD to both, we see that BE is equal to the sum of AD and DB. Now, £F being a diameter of the circle of which the outer segment is a part, BF is greater than BE ; therefore AC, CB are together greater than AD, DB. And, generally, of ail trianglts oh tht same bas€ and on the same side of it whkh hat>e tqttai vertical angles, the isosales triangle is that whieh has the greatest perimeter, and of the others that has the lesser perimeter which is further from being isosceles. The theorem of Serenus gives us the means of solving the following problem given in.Todhunter's Euclid, p 314. To find a point in the eirmmferenee of a given segment of a circle such that the straight Una which Jain the point to the extremities of the straight line on which tlu segment stands may ^ together equal to a given straight line (the length of which is of course subject to limits). Let A CB in the above figure be the given segment. Find, by bisecting AB at right angles, a point C on it such that ^ C is equal to CB. Then with centre C and radius CA or CB describe the segment of a ctccle AHB on the satne side of AB. Lastly, with ^4 or i' as centre and radius equal to the given straight line describe a circle. This circle will, if the given straight line be greater than AB and less than twice AC, meet the outer segment in two points, and if we join those points to the centre of the circle last drawn (whether A or B\ the joining straight lines will cut the inner segment in points satisfying the given condition. If the given straight line be eguai to twice AC, C is of course the required point. If the given straight line be greater than twice .^JC, there is no possible solution. ' ' ■■ ' ' Proposition 22, The opposite angles of qiutdrilaterah in circles are equal to two right angles. Let ABCD be a circle, and let ABCD be a quadrilateral in it; 1 say that the opposite ang^t.3 are equal to two right angles. Let AC, BD be joined. Then, since in any triangle the three angles are equal to two right angles, [1. 31] S» BOOK III [ill. *J, a3 the three angles CAB, ABC, BCA of the triangle ABC are equal to two right angles. But the angle CAB is equal to the angle BDC, for they are in the same segment BADC; [m. ii] and the angle ACS is equal to the angle ADB, for they are in the same segment ADCB', therefore the whole angle ADC is equal to the angles BAC, ACB. Let the angle ABC be added to each ; i ^ therefore the angles ABC, BAC, ACB are equal to the angles ABC, ADC. But the angles ABC, BAC, ACB are equal to two right angles ; therefore the angles ABC, ADC are also equal to two right angles. Similarly we can prove that the angles BAD, DCB are also equal to two right angles. Therefore etc, Q. E. D. As Todhunter remarks, the converse of this proposition is true and very important : if hvo opposite anglts of a quadrilaitral bt togeShtr equal to two right angin, a dnk may de (ircumsmbed about the quadrilateral. We can, by the method of in. 9, or by iv. 5, circumscribe 3 circle about the triangle ABC; and we can then prove, by reductio ad nbsurdum, that the circle passes through the fourth angular fioint D. • ,.15 . Proposition 23. On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side. For, if possible, on the same straight line AB let two similar and unequal segments of circles ACB, ADB be constructed on the same side ; ^-^ii^^^^^ ° let A CD be drawn through, and let CB, DB be joined. Then, since the segment ACB is similar to the segment ADB, III. »3, 24] PROPOSITIONS aa— 14 53 and similar segments of circles are those which admit equal angles, [m. Def. u] the angle ACB is equal to the angle ADB, the exterior to the interior : which is impossible. , [i. 16] Therefore etc. I. cannot be conatnicled, ti nwro^rrcu, the stmt phnae is in [. 7. Clavius and the other early editors point out that, while the words "on the same side " in the enunciation are necessary for Euclid's proof, it is equally true that neither can there be two similar and unequal segments on apposite sides of the same straight line ; this is at once made clear by causing one of the segments to revolve round the base till it is on the same side with the other. Simson observes with reason that, while Euclid in the following proposition, III. 24, thinks It necessary to dispose of the hypothesis that, if two similar segments on equal bases are applied to one another with the bases coincident, the segments cannot cut in any other jwint than the extremities of the base (since otherwise two circles would cut one another in more points than two), this remark is an equally necessary preliminary to iii. 23, in order that we may be justified in drawing the segments as being one inside the other- Sim son accordingly begins his proof of in. 23 thus : "Then, because the circle ACB cuts the circle ADB in the two points A, B, they cannot cut one another in any other point : One of the segments must therefore fall within the other. Let ACS fall within ADB and draw the straight line ACI), etc." Simson has also substituted "not coinciding with one another" for "unequal" in Euclid's enunciation. Then in i[i. 24 Simson leaves out the words referring to the hypothesis that the segment AEB when applied to the other CFD may be " otherwise placed as CGD" \ in fact, after stating that AS must coincide with CD, he merely adds words quoting the result of ni. aj : "Therefore, the straight line j4.^ coinciding with CD, the segment AEB must coincide with the segment CFD, and is therefore equal to it." Proposition 24. Similar segments of circles on equal straight lines are equal to one another. For let AEB, CFD be similar segments of circles on eqtial straight lines AB, CD ; s I say that the segment AEB is equal to the segment CFD. For, if the segment AEB be applied to CFD, and if the point A be placed on C and the straight line AB on CZ?, 54 BOOK in "" [hi. 34, *s the point B will also coincide with the point D, because AB\& equal to CD ; 10 and, ^^ coinciding with CD, ' • ; .• . ■ 1 . , ' •. !;'r(j the segment AEB will also coincide with CFD. ' r For, if the straight line AB coincide with CD but the segment AEB do not coincide with CFD, it will either fall within it, or outside it ; • i- IS or it will fall awry, as CGD, and a circle cuts a circle at more points than two : which is impossible. [m. 10] Therefore, if the straight line AB be applied to CD, the segment AEB will not fail to coincide with CFD also ; therefore it will coincide with it and will be equal to it. 20 Therefore etc, ' ' • • Q. E. D. rj . fftU awiy, TopoWd^fc, the same v/ord a& used in tfae Uke case in [. S. The word impHes that the applied figure will partly fall short of, and partly overlap, the Aguie to which it is applied^ Compare the note on the last proposition. I have put a semicolon instead of the comma which the Greek text has after "outside it," in order the better to indicate that the inference "and a circle cuts a. circle in more points than two " only refers to the third hypothesis that the applied segment is "otherwise placed (ifapaAXftfft) as CGD." The first two hypotheses are disposed of by a tacit reference to the preceding proposition in. 23, Proposition 25. Given a segment 0/ a circle, to describe the complete circle ofiuhich it is a segment. Let ABC be the given segment of a circle ; thus it is required to describe the complete circle belonging to the segment ABC, that is, of which it is a segment. For let AC he. bisected at D, let DB be drawn from the point D at right angles to AC, and let AB, be joined ; uj. as] PROPOSITIONS 24, 25 55 the angle ABD is then greater than, equal to, or less than the angle BAD. First let it be greater ; ' ' ' "^ and on the straight line BA, and at the point A on it, let the angle BAE be constructed equal to the angle ABD; let DB be drawn through to E, and let EC be joined. Then, since the angle ABE is equal to the angle BAE, the straight line EB is also equal to EA. [1. 6] And, since AD is equal to DC, and DE is common, ., . -< 1 the two sides AD, DE are equal to the two sides CD, DE respectively ; and the angle ADE is equal to the angle CDE, for each is right ; . ,, therefore the base AE is equal to the base C£. • But AE was proved equal to BE ; therefore BE is also equal to CE ; therefore the three straight lines AE, EB, EC are equal to one another. Therefore the circle drawn with centre E and distance one of the straight lines AE, EB, EC will also pass through the remaining points and will have been completed, [ni. 9] Therefore, given a segment of a circle, the complete circle has been described. And it is manifest that the segment ABC is less than a semicircle, because the centre E happens to be outside it. Similarly, even if. the angle ABD be equal to the angle BAD, AD being equal to each of the two BD, DC, the three straight lines DA, DB, DC will be equal to one another, D will be the centre of the completed circle, and ABC will clearly be a semicircle. 56 BOOK III ' [ill. 35, 3fi But, if the angle ABD be less than the angle BAD, and if we construct, on the straight line BA and at the point A on it, an angle equal to the angle ABD, the centre will fall on DB within the segment ABC, and the segment ABC will clearly be greater than a semi- circle. Therefore, given a segment of a circle, the complete circle has been described. Q. E. F. T. to deaciibe the complete circle^ itpatravaypd^pat tov kAxXov, Utenlly "Lo describe the circle tm fo it. * It will be remembered that Sim son takes first the case in which the angles ABD, BAD are equal to one another, and then takes the other two cases together, telling us to "produce BD, if necessary." This is a little shorter than Euclid's procedure, though Euclid does not repeat the proof of the first case in giving the third, but only refers to it as equally applicable. Campanus, Petetarius and others give the solution of this problem in which we take two chords not parallel and bisect each at rj^ht angles by straight lines, which must meet in the centre, since each contains the centre and they only intersect in one point. Clavius, Billingsley, Barrow and others give the rather simpler solution in which the two chords have one extremity common (cf. Euclid's proofs of lit. g, ro). This method De Morgan favours, and (as noted on in. i above) would make iii, i, this proposition, and IV, 5 all coroilaries of the theorem that " the line which bisects a chord perpendicularly must contain the centre," Mr H. M. Taylor practically adopts this order and method, though he finds the centre of a circle by means of any two non -parallel chords ; but he finds the ctntre of the circle of whkh a given art is a part (his proposition corresponding to in. 15) by bisecting at right angles first the base and then the chord joining one extremity of the base to the point in which the line bisecting the base at right angles meets the circumference of the segment. Under De Morgan's alternative the relation between Euclid in. i and the Porism to it would be reversed, and Euclid's notion of a Porism or corollary would have (o be considerably extended. If the problem is solved ^fter the manner of iv. 5, it is still desirable to state, as Euclid does, after proving AE, EB, EC to be all equal, that "the circle drawn with centre E and distance one of the straight lines AE, EB, EC will also pass through the remaining points of the segment" [ni. 9], in order to show that part of the circle described actually coincides with the given segment. This is not so clear if the centre is determined as the intersection of the straight lines bisecting at right angles chords which join pairs of four different points. Proposition 26. fn equal arcles equal angles stand on equal arcumferenees, wheUier they stand at tlie centres or at the circumferences. III. a6] PROPOSITIONS 25, 26 57 Let ABC, DEF be equal circles, and in them let there be equal angles, namely at the centres the angles BGCt EHF, and at the circumferences the angles BAC, EDF\ I say that the circumference BKC is equal to the circum- ference ELF. ■' 1,1 ■\ For let i?C, ^^ be joined. • . Now, since the circles ABC, DEFz.re equal, the radii are equal. Thus the two straight lines BG, GC are equal to the two straight lines EH, HF; and the angle at G is equal to the angle at H; therefore the base BC is equal to the base EF. [1. 4] And, since the angle at A is equal to the angle at D, the segment BAC is similar to the segment EDF; [hi. Def. 11] and they are upon equal straight lines. But similar segments of circles on equal straight lines are equal to one another ; [in. 34] therefore the segment BAC is equal to EDF. But the whole circle ABC is also equal to the whole circle DEF: therefore the circumference BKC which remains is equal to the circumference j£'Z/*'. ,.-.,... Therefore etc. Q, e. d. As in HI. 21, if Euclid's ptoof is to cover all cases, it requires us to take cognisance of " angles at the centre " which are equal w or greater than two ■right angles. Otherwise we must deal separately with the cases where the angle at the circumference is equal to or greater than a right angle. The case of an ebtust angle at the circumference can of course be reduced by means of ni. iz to the case of an acute angle at the circumference; and, in case the angle at the circumference is right, it is readily proved, by drawing the radii to the vertex of the angle and to the other extremities of the lines containing it, that the latter two radii are in a straight line, whence they make equal bases in the two circles as in Euclid's proof. SS BOOK III [lit. 36.17 Lordner has another way of dealing with the right angle or obtuse angte at the circumference. In either case, he says, "bisect them, and the halves of them are equal, and it can be proved, as above, that the arcs upon which these halves stand are equal, whence it follows that the arcs on which the given angles stand are equal." Proposition 27, /« egua^ circles angles standing on equal circumferences are equal ta one another, whether they stand at the centres or at the circumferences. For in equal circles ABC, DBF, on equal circumferences BC, EF, let the angles BGC, EHF stand at the centres G, H, and the angles BAC, EDF 3X the circumferences ; I say that the angle BGC is equal to the angle EHF, and the angle BAC is equal to the angle EDF. For, if the angle BGC is unequal to the angle EHF, one of thetn is greater. Let the angle BGC be greater : and on the straight line BG, and at the point G on it, let the angle BGK be constructed equal to the angle EHF. [1. as) Now equal angles stand on equal circumferences, when they are at the centres ; [m. a6] therefore the circumference BK is equal to the circum- ferencer EF. But EF is equal to BC ; therefore BK is also equal to BC, the less to the greater : which is impossible. Therefore the angle BGC is not unequal to the angle EHF; therefore it is equal to it HI. a7, »8] PROPOSITIONS 16—28 $9 And the angle at A is half of the angle BGC, and the angle at D half of the angle EHF\ [m. ao] therefore the angle at A is also equal to the angle at D. Therefore etc. Q. E. D. This proposition is the converse of the preceding one, and the remarks about the method of treating the different cases apply here also. Proposition 28. In egtial circles equal straight lines cut off equal circum- ferences, the greater equal to the greater and Ike less to tlie less, .■;, ■ - ..'1. ' <T ' • Let ABC, DEF be equal circles, and in the circles let AB, DE be equal straight lines cutting off ACS, DEE as greater circumferences and AGB, DHE as lesser; I say that the greater circumference ACB is equal to the greater circumference DFE, and the less circumference ^G^jff to DHE. For let the centres K, L of the circles be taken, and let ^ A", ^^, Z?Z., Z.^ be joined. . -, v, -i ^./ j- Now, since the circles are equal, . . the radii are also equal ; therefore the tjvo sides AK, KB are equal to the two sides DL, LE ; and the base AB is equal to the base DE ; therefore the angle A KB is equal to the angle DLE. [I. 8] But equal angles stand on equal circumferences, when they are at the centres ; [m. 26] therefore the circumference AGB is equal to DHE. 6ai « BOOK. Ill ";^•^ [HI. a8, J9 And the whole circle ABC is also equal to the whole circle DEF\ therefore the circumference ACB which remains is also equal to the circumference DFE which remains. Therefore etc. Q. E. D, Euclid's proof does not in terms cover the particular case in which the chord in one circle passes through its centre ; but indeed this was scarcely worth giving, as the proof can easily be supplied. Since the chord in one circle passes through its centre, the chord in the second circle must also be a diameter of that circle, for equal circles are those which have equal diameters, and all other chords in any circle are less than its diameter [in. 15]; hence the segments cut off in each circle are semicircles, and these must be equal because the circles are equal. Proposition 29. In equal circles equal circumferences are subtended by equal straight lines. Let ABC, DBF be equal circles, and in them let equal circumferences BGC, EHF be cut off; and let the straight lines BC, EF be joined ; I say that BC is equal to EF. For let the centres of the circles be taken, and let them be a: Z ; let BK, KC, EL, Z/^ be joined. Now, since the circumference BGC is equal to the circumference EHF, the angle BKC is also equal to the angle .£'Z/^ [m. 17] And, since the circles ABC, DEF are equal, the radii are also equal ; therefore the two sides BK^ KC are equal to the two sides EL, LF; and they contain equal angles ; therefore the base BC is equal to the base EF. [i. 4] Therefore etc. iti. 29—31] PROPOSITIONS *8— 31 61 The particular case of this converse of ill. 28 in which the given arcs are ares of semicircles is even easier than the corresponding case of in, 18 itself. The propositions in, z6 — 29 are of couise equaliy true if the same circle is taken instead of iwo equal circles. Proposition 30. To bisect a given circumference. Let ADB be the given circumference ; thus it is required to bisect the circumference ADB. Let AS h^ joined and bisected at C ; from the point C let CD be drawn o at right angles to the straight line AB, and let AJJ, DB be joined. Then, since ACb^ equal to CB, and CD is common, the two sides A C, CD are equal to the two sides BC, CD ; and the angle ACD is equal to the angle BCD, for each is right ; therefore the base AD is equal to the base DB. [1. 4] But equal straight lines cut off equal circumferences, the greater equal to the greater, and the less to the less ; [in- «8] and each of the circumferences AD, DB is less than a semicircle ; therefore the circumference AD is equal to the circum- ference DB, Therefore the given circumference has been bisected at the point D. Q. E. F. ; ' Proposition 31. - , ,, , •' In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle ; and further the angle of the greater segment is greater than a right angle, and the angle of the less segment less than a right angle. «» BOOK m [ill. 31 - Let ABCD be a circle, let BC be its diameter, and E its centre, and let BA, AC, AD, DC be joined ; I say that the angle BAC in the semicircle BAC is right, the angle ABC in the segment -^^C greater than the semicircle is less than a right angle, and the angle ADC in the segment ADC less than the semicircle is greater than a right angle. Let AE be joined, and let BA ,„ ... , ; be carried through to ^. , ., , . i Then, since BE is equal to EA^ the angle A BE is also equal to the angle BAE, [1. s] ; Again, since CE is equal to EA, ■..-^^i^ ■ /r* the angle ACE is also equal to the angle CAE. [1. s] Therefore the whole angle BAC is equal to the two angles ABC, ACB. But the angle EAC exterior to the triangle ABC is also equal to the two angles ABC, ACB ; [i. 31] therefore the angle BAC is also equal to the angle EAC; therefore each is right ; [1. Def. 10] therefore the angle BAC in the semicircle BAC is right. Next, since in the triangle ABC the two angles ABC, BAC are less than two right angles, [i. 17] and the angle BAC is a right angle, the angle ABC is less than a right angle ; and it is the angle in the segment ABC greater than the semicircle. Next, since ABCD is a quadrilateral in a circle, and the opposite angles of quadrilaterals in circles are equal to two right angles, [iil »] while the angle ABC is less than a right angle, therefore the angle ADC which remains is greater than a right angle ; and it is the angle in the segment ADC less than the semi- circle. HI. 31] PROPOSITION 31 63 I say further that the angle of the greater segment, namely that contained by the circumference ABC and the straight line AC, is greater than a right angle ; and the angle of the less segment, namely that contained by the circumference ADC and the straight line AC, is less than a right angle. This is at once manifest. For, since the angle contained by the straight lines BA, AC is right, the angle contained by the circumference ABC and the straight line AC is greater than a right angle. Again, since the angle contained by the straight lines AC, AFis right, the angle contained by the straight line CA and the circumference ADC is less than a right angle. Therefore etc. q. e. d. As already stated, this proposition is immediately deducible from in. 20 if that theorem Is extended so as to include the case where the segment is equal to or less than a semicircle, and where consequently the " angle at the centre" is equal to two right angles or greater than two right angles respectively. There are indications in Aristotle that the proof of the first part of the theorem in use before Euclid's time proceeded on different lines. Two passages of Aristotle refer to the proposition that the angle in a semicircle IS a right angle. The first passage is Anal. Fast 11. 11, 94 a 38: "Why is the angle in a semicircle a right arvgle? Or what makes it a right angle? (tivo! ovrtK op$ij;) Suppose ^4 to be a right angle, B half of two right angles, C the angle in a semicircle. Then B is the cause of j4, the right angle, being an attribute of C, the angle in the semicircle. For £ is equal to /*, and CtoB; for C is half of two right angles. Therefore it is in virtue of £ being half of two right angles that A is an attribute of C ; and the latter means the fact that the angle in a semicircle is right." Now this passage by itself would be consistent with a proof like Euclid's or the alterrmtive interpolated proof next to be mentioned. But the second passage throws a different light on the subject. This is Metaph, 1051 a 26 ; "Why is the angle in a semicircle a right angle invariably (dafloAou) ? Because, if there be three straight lines, two forming tkt base, and the third iet uf at right angles at its middle point, the fact is obvious by simple inspection to any one who knows the property referred to" {Ikuvo is the property that the angles of a triangle are together equal to two right angles, mentioned two lines before). That is to say, the an^le at the middle point of the circumference of the semicircle was taken and proved, by means of the two isosceles right-angled triangles, to be the sum of two angles each equal to one-fourth of the sum of the angles of the large triangle in the figure, or of two right angles; and the proof . ?. -'.t must have been completed by means of the theorem of lit. zi (that angles £t BOOK III [III. 31 in the same s^ment are equal), which Euclid's more general proof does not need. In the Greek texts before that of August there is an alternative proof that the angle BAC (in a semicircle) is right. August and Heiberg rel^;ate it to an Appendix. " Since the angle AEC is double of the angle BAE (for it is equal to the two interior and opposite angles), while the angle AEB b also double of the angle EAC, the angles AEB, AECatft double of the angle SAC. ■ ■ But the angles AEB, AEC are equal to two right angles J • ■• ' 1 therefore the angle BAC is right." Lardner gives a slightly different proof of the second part of the theorem. If ABC be a segment greater than a semicircle, draw the diameter AD^ and join CH, CA. Then, in the triangle ACD, the angle ACD is right (being the angle in a semicircle) ; therefore the angle ADC\s acute. But the angle ADC i^ equal to the angle ABC in the same segment ; therefore the angle ABC b acute. Euclid's references in this proposition to the angle of a s^ment greater or less than a semicircle respectively seem, like the part of 111. 16 relating to the angle 0/ a semicircle, to be a survival of ancient controversies and not to be put in deliberately as being an essential part of elementary geometry. Cf. the notes on 111, Def. 7 and in. 16. The corollary ordinarily attached to this proposition is omitted by Heibetg as an interpolation of date later than Theon. It is to this effect ; " From this it is manifest that, if one angle of a triangle be equal to the other two, the first angle is right because the exterior angle to it is also equal to the same angles, and if the adjacent angles be equal, they are right." No doubt the corollary is rightly suspected, because there is no necessity for it here, and the words oirip itti Siifai come before it, not after tt, as is usual with Euclid. But, on the other hand, as the fact stated does appear in the proof of 111. 31, the Porism would be a Porism after the usual type, and I do not quite follow Hei berg's argument that, "if Euclid had wished to add it, he ought to have placed It after 1. 3*." It has already been mentioned above (p. 44) that this proposition supplies us with an alternative construction for the problem in 111, 1 7 of drawing the two tangents to a circle from an external point. Two theorems of some historical interest which follow directly from in. 3r may be mentioned. The first is a lemma of Pappus on " the 14th problem " of the second Book of Apol- lonius' lost treatise on vcvVtit (Pappus vii. p. 811) and is to this effect. If a circle, as DEF, pass through D, the centre of a circle ABC, and if through F, the other point in which the line of centres meets the circle DEF, any straight line be drawn (and produced if necessary) meeting the circle DEF in E and the circle ABC in B, G, m- 3*. 3'] PROPOSITIONS 3t, %i «S then E is the middle point of £G. For, if UE be joined, the angle I>EJ^ (in a. semicircle) is a right angle [iii. 31] ; and DE, being at right angles to the chord BG of the circle A£C, also bisects it [m. 3]. The second is a proposition in the Zi^r Asiumpiarttm, attributed (no doubt erroneously as regards much of it) to Archimedes, which has reached us through the Arabic (Archimedes, ed, Heiberg, 11. pp. 52© — -5 21)' If two chords AB, CD iit a circle infersicl at right angles in a point O, thtn the sum of the squares on AG, BO, CO, DO is equal to the square on the diameter. For draw the diameter CE, and join AC, CB, AD, BE. , jj Then the angle CAO is equal to the angle CES. (This follows, in the first figure, from iii. 31 and, in the second, from 1. 13 and ill. 22.) Also the angle COA, being right, is equal to the angle CBE which, being the angle in a semicircle, is also right [iii. 31]. Therefore the triangles AOC, EBCh^ve two angles equal respectively; whence the third angles A CO, £CJ5 are equal. (In the second figure the angle A CO is, by i. 13 and 111. aa, equal to the angle ABD, and therefore the angles ABD, ECB are equal) Therefore, in both figures, the arcs AD, BE, and consequently the chords AD, BE subtended by them, are equal. [111. 36, 29] Now the squares on AO, DO are equal to the square on AD\\. 47}, that is, to the square on BE. And the squares on CO, £0 Mt equal to the square on BC. Therefore, by addition, the squares on AO, BO, CO, DO are equal to the squares on EB, BC, i.e. to the square on CE, [1. 47J Proposition 32. • ' If a straight line touch a circle, and from the point of contact there be drawn across, in the circle, a straight line cutting the circle, the angles which it makes with the tangent will be equal to the angles in the alternate segments of the circle. For let a straight line EF touch the circle A BCD at the point B, and from the point B let there be drawn across, in the circle ABCD, a straight line BD cutting it ; I say that the angles which BD makes with the tangent EF will be equal to the angles in the alternate segments of the M BOOK III [til. 31 circle, that is, that the angle FBD is equal to the angle constructed in the segment BAD, and the angle EBD is equal to the angle constructed iii the segment DCB. For let BA be drawn from B at right angles to EF, let a point C be taken at random on the circumference BD, and let AD, DC, CB be joined. Then, since a straight line EF touches the circle A BCD at B, and BA has been drawn from the point of contact at right angles to the tangent, the centre of the circle ABCD is on BA. [m. 19] Therefore BA is a diameter of the circle ABCD ; therefore the angle ADB, being an angle in a semicircle, is right. [ill. 31] Therefore the remaining angles BAD, ABD are equal to one right angle. [1. 32] But the angle ABF is also right ; therefore the angle ABF is equal to the angles BAD, ABD. Let the angle ABD be subtracted from each ; therefore the angle DBF which remains is equal to the angle BAD in the alternate segment of the circle. Next, since ABCD is a quadrilateral in a circle, its opposite angles are equal to two right angles. [iii. a»] But the angles DBF, DBF are also equal to two right angles ; therefore the angles DBF, DBF are equal to the angles BAD, BCD, of which the angle BAD was proved equal to the angle DBF; therefore the angle DBF which remains is equal to the angle DCB in the alternate segment DCB of the circle. Therefore etc. q. e. d. The converse of this theorem is true, namely that, If a straight iine drawn through one txtraniiy of a chord of a circle make with that chord angles equal respectively to the angles in the alternate segments of the ctrde, the straight line so drawn touches the circle. ""■ ih 33] PROPOSITIONS 33, 33 67 This can, as Camerer and Tod hunter remark, be proved indirectly ; or we may prove it, with Clavius, directly. Let BD be the given chord, and let £F be drawn through B so that it makes with BD angles equal to the angles in the alternate segments of the circle respectively. Let BA be the diameter through B, and let C be any point on the circumference of the segment DCB which does not contain A. Job A£>f DC, CB. Then, since, by hypothesis, the angle FBD is equal to the angle BAD, let the angle ABI> be added to both; therefore the angle ABF\i equal to the angles AJSD, BAD. But the angle BDA, being the angle in a semicircle, is a right angle ; therefore the remaining angles ABD, BAD in the triangle ABD are equal to a right angle. Therefore the angle ABFi^ right ; hence, since BA is the diameter through B, £i^ touches the circle at A [cii. 16, Por,] Pappus assumes in one place (iv, p. 196) the consequence of this proposition that, If two eircks touch, any straight line drawn through the point of (oniad and terminated by both cireiei mis off segments in each which are respediwly similar. Pappus also shows how to prove this (vii, p, 8i6) by drawing the cominon tangent at the point of contact and using thb proposition. Proposition 33. On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilineal angle. Let AB be the given straight line, and the angle at C the given rectilineal angle; . . rv - thus it is required to describe on the given straight line AB a segment of a circle ad- mitting an angle equal to the angle at C. The angle at C is then acute, or right, or obtuse. First let it be acute, and, as in the first figure, on the straight line AB, and at the point A, let the angle BAD be constructed equal to the angle at C ; therefore the angle BAD is also acute. Let AE h& drawn at right angles to DA, let AB be \ «» rf BOOK m [""■ iS bisected at /^, let FG be drawn from the point F at right angles to AB, and let GB be joined. Then, since Af is equal to fB, and FG is common, the two sides AF, FG are equal to the two sides BF, FG ; and the angle AFG is equal to the angle BFG ; therefore the base AG is equal to the base BG. [i. 4] Therefore the circle described with centre G and distance GA will pass through B also. Let it be drawn, and let it be ABE ; let EB be joined. Now, since AD is drawn from A, the extremity of the diameter AE, at right angles to AE, therefore AD touches the circle ABE. \\\\. 16, Por.] Since then a straight line AD touches the circle ABE, and from the point of contact at A a straight line AB is drawn across in the circle ABE, the angle DAB is equal to the angle AEB in the alternate segment of the circle. [ni. 31] But the angle DAB is equal to the angle at C; therefore the angle at C is also equal to the angle AEB. Therefore on the given straight line AB the segment AEB of a circle has been described admitting the angle AEB equal to the given angle, the angle at C 1 ■ Next let the angle at C be right ; and let it be again required to describe on AB a segment of a circle admitting an angle equal to the right angle at C. Let the angle BAD be constructed equal to the right angle at C, as is the case in the second figure ; 111. 33] PROPOSITION 33 69 let j4B h& bisected at /^, and with centre /^ and distance either I^A or /^B let the circle AEB be described. Therefore the straight line AD touches the circle ABE, because the angle at A is right. [m. i6j Por.] And the angle BAD is equal to the angle in the segment AEB, for the latter too is itself a right angle, being an angle in a semicircle. [ni, 31) But the angle BAD is also equal to the angle at C. Therefore the angle AEB is also equal to the angle at C. Therefore again the segment AEB of a circle has been described on AB admitting an angle equal to the angle at C. Next, let the angle at C be obtuse ; .i»' "E . 'I and on the straight line AB, and at the point A, let the angle BAD be constructed equal to it, as is the case in the third figure ; let AE be drawn at right angles to AD, let AB be again bisected at F, let FG be drawn at right angles to AB, and let GB be joined. Then, since AF is again equal to FB, and FG is common, the two sides AF, FG are equal to the two sides BF, FG ; and the angle AFG is equal to the angle BFG ; therefore the base AG is equal to the base BG. [i- 4] Therefore the circle described with centre G and distance GA will pass through B also ; let it so pass, as AEB. Now, since AD is drawn at right angles to the diameter AE from its extremity, AD touches the circle AEB. [m. 16, Por.] And AB has been drawn across from the point of contact at W ; therefore the angle BAD is equal to the angle constructed in the alternate segment AHB of the circle. [m. 31] f0 BOOK in ' ["'33.34 But the angle BAD is equal to the angle at C Therefore the angle in the segment A MB is also equal to the angle at C: Therefore on the given straight line AB the segment AHB of a circle has been described admitting an angle equal to the angle at C. Q, E, F. Simson remarks truly that the first and third cases, those namely in which the given angle is acute and obtuse respectively, have exactly the same construction and demonstration, so that there is no advantage in repeating them. Accordingly he deals with the cases as one, merely drawing two different figures. It is also true, as Simson says, that the demonstration of the second case in which the gi^-en angle is a right angle " is done in a round- about way," whereas, as Clavius showed, the problem can be more easily solved by merely bisecting AB and describing a semicircle on it. A glance at Euclid's figure and proof will however show a more curious fact, namely that he does not, in the proof of the second case, use the angle in the alternate stgmint, as he does in the other two cases. He might have done so after proving that AD touches the circle; this would only have required his point .£ to be placed on the side of AB opposite to D. Instead of this, he uses III. 31, and proves that the angle AEB is equal to the angle C, because the former is an angle in a sanicirde, and is therefore a right angle as C is. The difference of procedure is no doubt owing to the fact that he has not, in III. 31, distinguished the case in which the cutting and touching straight lines are at right angles, i.e. in which the two alternate segments are semicircles. To prove this case would also have required in. 31, so that nothing would have been gained by stating it separately in in. 32 and then quoting the result as part of 111. 32, instead of referring directly to in. 31. It is assumed in Euclid's proof of the first and third cases that AE and FG will meet; but of course there is no difficulty in satisfying ourselves of this. J Proposition 34. "" " " From a given circle to cut off a segment admitting an angle tqual to a given rectilineal angle. Let ABC be the given circle, and the angle at D the given rectilineal angle ; thus it is required to cut off from the circle ABC a segment admitting an angle equal to the given rectilineal angle, the angle at D. Let EF\i^ drawn touching ABC at the point B, and on the straight line FB, and at the point B on it, let the angle FBC be constructed equal to the angle at D. [1. 23] ' . Then, since a straight line EF touches the circle ABC^ "■•34. 3S] PROPOSITIONS 33-35 ji and BC has been drawn across from the point of contact at^, the angle FBC is equal to the angle constructed in the alternate segment BAC, [iti. 37] But the angle FBC is equal to the angle at D ; therefore the angle in the segment BAC is equal to the angle at D. Therefore from the given circle ABC the segment BAC, has been cut off" admitting an angle equal to the given recti- lineal angle, the angle at D. Q. E, F. An alternative construction here would be to make an "angle at the centre " {in the extended sense, if necessary) double of the given angle ; and, if the given angle is right, it is only necessary to draw a diameter of the circle. Proposition 35. Jf in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. For in the circle ABCD let the two straight lines AC, BD cut one another at the point E ; I say that the rectangle contained hy AB, EC is equal to the rectangle contained by DE, EB. If now AC, BD are through the centre, so that E is the centre of the circle ABCD, it is manifest that, AE, EC, DE, EB being equal, the rectangle contained by AE, EC is also equal to the rectangle contained by DE, EB. :ja /; BOOK in [hi. 35 Next let AC, DB not be through the centre ; let the centre of ABCD be taken, and let it be F\ from F let FG, FH be drawn perpen- dicular to the straight lines AC, DB, and let FB, FC, FE be joined. Then, since a straight line GF through the centre cuts a straight line AC not through the centre at right angles, it also bisects it ; [in. 3] therefore AG is equal to GC. Since, then, the straight line AC has been cut into equal parts at G and into unequal parts at E, the rectangle contained by AE, EC together with the square on EG is equal to the square on GC ; [11. 5] Let the square on GF be added ; therefore the rectangle AE, EC together with the squares on GE, GF is equal to the squares on CG, GF. But the square on FE is equal to the squares on EG, GF, and the square on FC Is equal to the squares on CG, GF\ ['. 47l therefore the rectangle AE, EC together with the square on FE is equal to the square on FC. And FC is equal to FB ; therefore the rectangle AE, EC together with the square on EF is equal to the square on FB. For the same reason, also, the rectangle DE, EB together with the square on FE is equal to the square on FB. But the rectangle AE, EC together with the square on FE was also proved equal to the square on FB ; therefore the rectangle AE, EC together with the square on FE is equal to the rectangle DE, EB together with the square on FE. Let the square on FE be subtracted from each ; therefore the rectangle contained by AE, EC which remains is equal to the rectangle contained by DE, EB, Therefore etc. III. 35, 36] PROPOSITIONS 35. 36 m In addition to the two cases in Euclid's text, Simson (following Campanus) gi.ves two intermediate cases, namely (i) that in which one chord passes through the centre and bisects the other which does not pass through the centre at right angles, and (a) that in which one passes through the centre and cuts the other which does not pass through the centre but not at right angles Simson then reduces Euclid's second case, the most general one, to the second of the two intermediate cases by drawing the diameter through £. His note is as follows : "As the 25th and 33rd propositions are divided into more cases, so this 35th is divided into fewer cases than are necessary. Nor can it be supposed that Euclid omitted them because they are easy ; as he has given the case which by far is the easiest of them all, viz, that in which both the straight lines pass through the centre ; And in the following proposition he separately demonstrates the case in which the straight line passes through the centre, and that in which it does not pass through the centre: So that it seems Theon, or some other, has thought them too long to insert : But cases that require different demonstrations should not be left out in the Elements, as was before taken notice of: These cases are in the translation from the Arabic and are now put into the text." Notwithstanding the ingenuity of the argument based on the separate mention by Euclid of the simplest case of all, I think the conclusion that Euclid himself gave /our cases is unsafe ; in fact, in giving the simplest and most difficult cases only, he seems to be following quite consistently his habit of avoiding Aw ^reai multiplicity of cases, while not ignoring their existence. The deduction from the next proposition (in, 36) which Simson, following Clavius and others, gives as a corollary to it, namely that, IJ from any point without a drcU then be drawn two straight tines cutting it, the rectangles contained by the whole lines and the parts of them without the circle are equal t& one another, can of course be combined with ill. 35 in one enunciation. As remarked by Todhunter, a large portion of the proofs of in, 35, 36 amounts to proving the proposition, If any point be taken on the bast, or the base produced, of an isosceles triangle, the rectangle contained by the segments of the base (i.e. the respective distances of the ends of the base from the point) is equal to the difference betiveen the square on the straight line joining the point to the vertex and the square on one of the equal sides of the triangle. This is of course an immediate consequence of 1, 47 combined with ii. 5 or 11. 6, The converse of in, 35 and Simson's corollary to lu. 36 may be stated thus. If two straight lines AB, CXi, produced if necessary, intersect at O, and if the rectangle AO, OB be equal to the rectangle CO, OD, the circumference of a circle will pass through the four points A, B, C, D. The proof is indirect. We describe a circle through three of the points, as A, B, C (by the method used in Euclid's proofs of tii. 9, 10), and then we prove, by the aid of in. 35 and the corollary to in. 36, that the circle cannot but pass through D also, » . .ii\ .<i A TM •.' ^. ■ 5 ■■- . 'fi v'* r •(' I Proposition 36. If a point be taken outside a circle and from it there fall on the circle two straight lines, and if one of tliem cut the circle and the other touch ii, the rectangle contained by the whole of the straight line which cuts the circle and the straight f4 BOOK, in ■: [111.36 line intercepted on it outside between the point and the convex circumference will be equal to the square on tke tangent. For let a point D be taken outside the circle ABC, and from D let the two straight lines DC A, DB fall on the circle ABC; let DCA cut the circle ABC and let BD touch it ; I say that the rectangle contained by AD, DC is equal to the square on DB. Then DCA is either through the centre or not through the centre. First let it be through the centre, and let F be the centre of the circle ABC; let FB be joined ; therefore the angle FBD is right. [m. 18] And, since AC has been bisected at F, and CD is added to it, the rectangle AD, DC together with the square on FC is equal to the square on FD. [11. 6] But FC is equal to FB ; therefore the rectangle AD, DC together with the square on FB is equal to the square on FD. And the squares on FB, BD are equal to the square on FD ; [i. 47] therefore the rectangle AD, DC together with the square on FB is equal to the squares on FB, BD. Let the square on FB be subtracted from each ; therefore the rectangle AD, DC which remains is equal to the square on the tangent DB. Again, let DCA not be through the centre of the circle ABC; let the centre E be taken, and from E let EF be drawn perpendicular x.o AC; let EB, EC, ED be joined. Then the angle EBD is right. [ill. 18] And, since a straight line EF through the centre cuts a straight line AC not through the centre at right angles, it also bisects it ; [in. 3] therefore AF is equal to FC. jii. 36. 37] PROPOSITIONS 36, 37 ?S Now, since the straight line ^Chas been bisected at the point F, and CD is added to it, the rectangle contained by AD, DC together with the square on FC is equal to the square on FD. [11, 6] Let the square on FE be added to each ; therefore the rectangle AD, DC together with the squares on CF, FE is equal to the squares on FD, FE. But the square on EC is equal to the squares on CF, FE, for the angle EEC is right ; [1. 47] and the square on ED is equal to the squares on DF, FE ; therefore the rectangle AD, DC together with the square on EC is equal to the square on ED. And EC is equal to EB ; therefore the rectangle AD^ DC together with the square on EB is equal to the square on ED. But the squares on EB, BD are equal to the square on ED, for the angle EBD is right ; [i. 47] therefore the rectangle AD, DC together with the square on EB is equal to the squares on EB, BD. Let the square on EB be subtracted from each ; therefore the rectangle AD, DC which remains is equal to the square on DB. Therefore etc, q. e. d, Cf. note on the preceding proposition. Observe that, whereas it would be natural with us to prove first that, if A is an external point, and two straight lines AEB, AFC cut the circle in E, B and F, C respectively, the rectangle BA, AE h equal to the rectangle CA, AF, and thence ihat, the tangent from A being a straight tine likt AEB in its limiting position when E and B coincide, either rectangle is equal to the square on the tangent (cf. Mr H. M. Taylor, p, 153), Euclid and the Greek geometers generally did not allow themselves to infer the truth of a proposition in a limiting case directly from the general case including it, but preferred a separate proof of the limiting case (cf. Apollonius of Perga, p. 40, 139 — 140). This accounts for the form of 11 r. 36. PRorosiTioN 37, If a point be taken outside a iireie and from the point there fall on the circle two straight lines, if one of them cut the circle, and the other fall on it, and if further the rect- angle contained by tlte whole of the straight line which cuts ^1 ,. BOOK III [hi, 37 Ike circle and the straight line intercepted on it outside between the point and the convex circumference be equal to the square on the straight line which falls on the circle, the straight line which falls on it will touch the circle. For let a point D be taken outside the circle ABC; from D let the two straight lines DCA, DB fall on the circle ACB; let DC A cut the circle and DB fall on it ; and let the rectangle AD, DC be equal to the square on DB. I say that DB touches the circle ABC. For let DE be drawn touching ABC ; let the centre of the circle ABC be taken, and let it be F\ let FE, FB, FD be joined. Thus the angle FED is right. [m. i8] ' ' Now, since DE touches the circle ABC, and DC A cuts it, the rectangle AD, DC is equal to the square on DE, [m. 36] But the rectangle AD, DC vt^.s also equal to the square onDB; therefore the square on DE is equal to the square on DB ; therefore DE is equal to DB. ■ And FE is equal to FB ; 1 therefore the two sides DE, EF are equal to the two sides DB, BF; and FD is the common base of the triangles ; therefore the angle DEF is equal to the angle DBF. [l 8] But the angle DEF is right ; therefore the angle DBF is also right. And FB produced is a diameter ; and the straight line drawn at right angles to the diameter of a circle, from its extremity, toucnes the circle ; [iir. 16, For.] therefore DB touches the circle. Similarly this can be proved to be the case even if the centre be on ^C. 1 Therefore etc. • <- q. e. d. ' De Morgan observes that there is here the same defect as in i. 48, i.e. an apparent avoidance of indirect demonstration by drawing the tangent DE on Iir, 37] PROPOSITION 37 ft the 0(^)OSite side of DF from UB. The case is similar to the appartnily direct proof which Campanus gave. He drew the straight line from D passing through the centre, and then (without drawing a second tangent) proved by the aid of n. 6 that the square on DP is equal to the sum of the squares on DB, BF\ whence {by t. 48) the angle DBF is a right angle. But this proof uses I. 48, the very proposition to which De Morgan's original remark relates. The undisguised indirect proof is easy. If DB does not touch the circle, it must cut it if produced, and it follows that the square on DB must be equal to the rectangle contained by DB and a longer line ; which is absurd. ;ti ' t BOOK IV. DEFINITIONS. I. A rectilineal Bgure is said to be inscribed in a rectilineal figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed. i. Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed. 3. A rectilineal figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle. 4. A rectilineal figure is said to be circumscribed about a circle, when each side of the circumscribed figure touches the circumference of the circle. 5. Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed. 6. A circle is said to be circumscribed about a figure when the circumference of the circle passes through each angle of the figure about which it is circumscribed. 7. A straight line is said to be fitted into a circle when its extremities are on the circumference of the circle. Definitions 1—7. ■ I api>end, as usual, the Greek text of the definitions. I, ^xi^jxa (v$vypafjLfiov tU (T^/lii tddvypafiftov tyypdifntTdat AryCTCU, orAy ttaimj TiSv Tol fyypa^fttKOu o^jtaTOt ymyitar iiida-rft wXtvpa^ rou, tit S iyypa^rTtLif airnfrai. IV, DEFF-. a— 7] DEFINITIONS 1—7 n 5, 2;^/xa lit^uypofifiav tl^ kvkXov Iffpo^^vStui Xtycrac, &rav cffairn^ ytaviti ToS tffpat^jiivav aiTTTfrat T^s tow kvicXod wtpi^tpiia.%. tXcitu tdS wtpiypa^ftivov i^Tmjrai riji tou kvkXou vtpi^tptiat. 5. KvitXot j4 (tt cr;(i7/ta o/UKOff lyypd()>i<rBai \iyirat, Srav 1) Tov kvkXou 6. KujfXo^ M ir^pl vx^fjta TTfpiypd^trSai Xrycrac, Jrav 17 tov ^icXov iripi^ipiia iitwmj^ 'yw»'(a¥ tou, wtpi S vtpiyp^^Tat, aTmjrtii. 7. Eutftra (ft KuKXer Jra/ijufffO'dai Xrytrai, oral' tu irc^aTa avr^f JTrt t^s wtptf^ptuL^ ^ Toi) fcuxXou, In the Rrst two definitions an English translation, if il is to be clear, must depart slightly from the exact words used in the Greek, where "each side" of one figure is said to pass through " each angle " of another, or " each angle " {I.e. angular point) of one ties on " each side " of another {Udimi rktvpd, iiiaiTD/ ytiiyia). It is also necessary, in the five definitions 1, 1, 3, 5 and 6, to translate the same Greek word aTrr^Toi in three different ways. It was observed on tit. Def. 2 that the usual meaning of arrurSai in Euclid is to metf, in contra- distinction to li^Trrt<r9ai, which means to fau^A. Exceptionally, as in Def. 5, iimaSoi has the meaning of iaucA. But two new meanings of the word appear, the first being to /ie on, zs in DefT. i atid 3, the second to pass through, as in DefT. 3 and 6; "each angle" lies on (airriTat) a side or on a circle, and " each side," or a circle, passes through (aurrrai) an angle or " each angle," The first meaning of lying an is exemplified in the phrase of Pappus af ctoi tJ a)\pM.w $(cr<( hi%op^lyrrp (Mtiat, "will lie on a Straight line given in position"; the meaning of passing through seems to be much rarer (I have not seen it in Archimedes or Pappus), but, as pointed out on itt. Def. 2, Aristotle uses the compound l^-miahai. in this sense. Simson proposed to read i^imjrat in the case (Def. 5) where an-njTot means touches. He made the like suggestion as regards the Greek text of ttl. II, 12, ij, 18, 19; in the first four of these cases there seems to be ms. authority for the compound verb, and in the fifth He! berg adopts Slmson's correction. ,[■ <Mn-' .'1 », . ! • «! ■'«';' . I- .J ^ •,.•?: ,.. .^ <{".<■ -'•to •y..\:z ,.T; n ' .•1 BOOK IV. PROPOSITIONS PrOI'OSITION I. fn/o a £iven circle to fit a straight line equal to a given straight line which is not greater than the diameter of the circle. Let ABC be the given circle, and D the given straight line not greater than the diameter of the circle ; thus it is required to fit into the circle ABC a straight line equal to the straight line D. Let a diameter BC of the circle ABC be drawn. Then, if BC is equal to D, that which was enjoined will have been done ; for BC has been fitted into the circle ABC equal to the straight line D. But, if BC is greater than D, let CE be made equal to D, and with centre C and distance CE let the circle EAR be described ; let CA be joined. Then, since the point C is the centre of the circle EAF, CA is equal to CE. But CE is equal to /? ; therefore D is also equal to CA. Therefore into the given circle ABC there has been fitted CA equal to the given straight line D. IV. I, a] PROPOSITIONS i, i $i or this problem as it stands there are of course an infinite number of solutions; and, if a particular point be chosen as one extremity of the chord to be "fitted in," there are two solutions. More difficult cases of "fitting into " a circle a chord of given length are arrived at by adding some further condition, e.g. (i) that the chord is to be parallel to a given straight line, or (2) that the chord, produced if necessary, shall pass through a given point. The former problem is solved by Pappus (in. p. rja); instead of drawing the chord as a tangent to a circle concentric with the given circle and having as radius a straight line the square on which is equal to the difference between the squares on the radius of the given circle and on half the given length, he merely draws the diameter of the circle which is parallel to the given direction, measures from the centre along it in each direction a length equal to half the given length, and then draws, on one side of the diameter, perpendiculars to it through the two points so determined. The second problem of drawing a chord of given length, being less than the diameter of the circle, and passing through a given point, is more important as having been one of the problems discussed by Apollonius in his work entitled vnxriit, now lost. Pappus states the problem thus (vii. p. 670): "A circle being given in position, to fit into it a straight line given in magnitude and verging (vtvoixrac) towards a given (point)." To do this we have only to place any chord HK in the given circle (with centre O) equal to the given length, take Z the middle point of it, with O as centre and OL as radius describe a circle, and lastly through the given point C draw a tangent to this circle meeting the given circle in j4, B, AB is then one of two chords which can be drawn satbfying the given conditions, if C is outside the inner circle ; if C\%Bn the inner circle^ there is one solution only ; and, if C is within the inner circle, there is no solution. Thus, if C is within the outer (given) circle, besides the condition that the given length must not be greater than the diameter of the circle, there is another necessary condition of the possibility of a solution, viz. that the given length must not be Itss than double of the straight line the square on which is equal to the difference between the squares (i) on the radius of the given circle and (2) on the distance between its centre and the given point. Proposition 2. In a given circle to inscribe a triangle equiangular with a given triangle. Let ABC be the given circle, and DBF the given triangle ; thus it is required to inscribe in the circle ABC a triangle equiangular with the triangle DEF. Let GH\x. drawn touching the circle ABC at A [m. i6,Por.]; 8a BOOK TV • [iT. » on the straight line AH, and at the point A on it, let the angle HAC be constructed eoual to the angle DEF, and on the straight line AG, and at the point A on it, let the angle GAB be constructed equal to the angle DFE ; let BC be joined. Then, since a straight line AH touches the circle ABC, and from the point of contact at A the straight line ^C is drawn across in the circle, therefore the angle HA C is equal to the angle ABC in the alternate segment of the circle. fin. 3*] But the angle HA C is equal to the angle DEF ; therefore the angle ABC is also equal to the angle DEF. For the same reason the angle ACB is also equal to the angle DFE ; therefore the remaining angle BAC is also equal to the remaining angle EDF. [i. 3*] Therefore in the given circle there has been inscribed a triangle equiangular with the given triangle. q. e. f. Here again, since any point on the circle niay be taken as an angular point of the triangle, there are an infinite number of solutions. Even when a particular point has been chosen to form one angular point, the required triangle may be constructed in six ways. For any one of the three angles may be placed at the point ; and, whichever is placed there, the positions of the two others relatively to it may be interchanged. The sides of the triangle will, in all the different solutions, be of the same length respectively ; only their relative positions will be different This problem can of course be reduced (as it was by Borelli) to nt. 34, namely the problem of cutting off from a given circle a segment containing an angle equal to a given angle. It can also be solved by the alternative method applicable to ni. 34 of drawing " angles at the centre " equal to double the angles of the given triangle respectively ; and by this method we can easily solve this problem, or \\\. 34, with the further condition that one aide of the IV. 3, 3] PROPOSITIONS a, 3 Sj required triangle, or the base of the required segment, respectively, shall be parallel to a given straight line. As a particular case, we can, by the method of this proposition, describe an tguilaterai triangle in any circle after we have first constructed any equilateral triangle by the aid of i. i. The possibility of this is assumed in IV. t6. It is of course equivalent to dividing the circumference of a circle into I Arte equttl parti. As De Morgan says, the idea of dividing a revolution into equal parts should be kept prominent in considering Book iv. ; this aspect of the construction of regular polygons is obvious enough, and the reason why the division of the circle into fh-et equal parts is not given by Euclid is that it happens to be as easy to divide the circle into three parts which are in the ratio of the angles of any triangle as to divide it into three equal parts. Proposition 3. About a given circle to circumscribe a triangle equiangular with a given triangle. Let ABC be the given circle, and DEF the given triangle ; 1 thus it is required to circumscribe about the circle ABC a triangle equiangular with the triangle DEF. Let EF be produced in both directions to the points G, H, let the centre K of the circle ABC be taken [in. r], and let 10 the straight line KB be drawn across at random ; on the straight line KB^ and at the point K on it, let the angle SKA be constructed equal to the angle DEG, and the angle BKC equal to the angle DFH ; [i. J3] and through the points A, B, C let LAM, MEN, NCL be 15 drawn touching the circle ABC. [in. 16, For] Now, since LM, MN, NL touch the circle ABC at the points A, B, C, and KA, KB, KC have been joined from the centre K to the points A, B, C, S| - BOOK IV ' itr>^ ao therefore the angles at the points A, B, C are right. [iii. i8] And, since the four angles of the quadrilateral AMBK are equal to four right angles, inasnnuch as AMBK is in fact divisible into two triangles, and the angles KAM, KBM are right, 25 therefore the remaining angles A KB, A MB are equal to two right angles. But the angles DEG, DEF are also equal to two right angles ; [1. 13] therefore the angles A KB, A MB are equal to the angles 30 DEG, DEF, of which the angle A KB is equal to the angle DEG ; therefore the angle AMB which remains is equal to the angle Z?^/^ which remains, Similarly it can be proved that the angle LNB is also 3S equal to the angle DFE \ therefore the remaining angle MLN is equal to the angle EDF. \\. 33] Therefore the triangle LMN is equiangular with the triangle DEF; and it has been circumscribed about the 40 circle ABC. Therefore about a given circle there has been circum- scribed a triangle equiangular with the given triangle. Q, E. F. ii». at raodom, Uterslly " ax it ma; chance," in trt/x"- The same etpression is used in ][]. I and commonly. 11. Is In fact dlviaible, lol SuupttTai, literally " is actually divided." The remarks as to the number of ways in which Prop, a can be solved apply here also. Euclid leaves us to satisfy ourselves that the three tangents }t>t7i meet and form a triangle. This follows easily from the fact that each of the artgles A^B, BKC, CKA is less than two right angles. The first two are so by construction, being the supplements of two angles of the given triangle re- spectively, and, since ail three angles round K are together equal to four right angles, it follows that the third, the angle AKC, is equal to the sum of the two angles E, Foi the triangle, i.e. to the supplement of the angle D, and is therefore less than two right angles. Peletarius and Borelli gave an alternative solution, flrst inscribing a triangle equiangular to the given triangle, by iv. 2, and then drawing tangents to the circle parallel to the sides of the inscribed triangle respectively. This method will of course give two solutions, since two tangents can be drawn parallel to each of the sides of the inscribed triangle. If the three pairs of parallel tangents be drawn and produced far enough, IV. 3, 4] PROPOSITIONS s, 4 »S they will form ^I'^Af triangles, two of which are the triangles ctrcumKribed to the circle in the manner required in the proposition. The other six triangles are so related to the circle that the circle touches two of the sides in each produced, i.e. the circle is an escribed circle to each of the six triangles. Proposition 4. In a given triangle to inscribe a circle. Let ABC be the given triangle ; thus it is required to inscribe a circle in the triangle ABC. Let the angles ABC, ACB S be bisected by the straight Hnes BD, CD [1. 9], and let these meet one another at the point D ; from D let DE, DF, DG be drawn perpendicular tothestraight 10 lines AB, BC, CA. Now, since the angle ABD is equal to the angle CBD, and the right angle BED is also equal to the right angle BFD, IS EBD, FBD are two triangles having two angles equal to two angles and one side equal to one side, namely that subtending one of the equal angles, which is BD common to the triangles ; therefore they will also have the remaining sides equal to 20 the remaining sides ; [i. 36] therefore DE is equal to DF, For the same reason DG is also equal to DF. Therefore the three straight lines DE, DF, DG are ec;ual as to one another ; therefore the circle described with centre D and distance one of the straight lines DE, DF, DG will pass also through the remaining points, and will touch the straight lines AB, BC, CA, because the angles at the points E, F, G 30 are right. For, if it cuts them, the straight line drawn at right angles to the diameter of the circle from its extremity will be found to fall within the circle : which was proved absurd ; [in. 16] ■9$ . BOOK IV [iv. 4 therefore the circle described with centre D and distance 35 one of the straight lines £>£, BF, DG will not cut the straight lines AB, BC, CA ; therefore it will touch them, and will be the circle inscribed in the triangle ABC. ["v. Def. s] Let it be inscribed, as FGE. 4° Therefore in the given triangle ABC the circle EFG has been inscribed. . - . ■.. ,. , , y. t,. r. i6, 34. and distance one of the (straight lines D)E, (D)F, (D)G. The wonte «nd letters here shown in brarkets are put in to fill out tbe ralher carelcK hngiiige o( ihe Greek- Hete and in several other places in Book IV. Euclid says lilemliy "and w I ih distance one of the (points) E,F,G" (lai Juunt^tan M rwc E, Z, H) and the like. In one case (1 v. 13) he actually has " with distance one of the pttintt G, If, IC, L, M" {tiuHmaTi irl rSr H, ©, K, A, H irit;i(lup). Heiberg notes" Craecam loculionem satis miram et negligenlem," but, in view of its frequent occurrence in good M3S., does not venture to correct it. Euclid does not think it necessary to prove that ££>, CD will meet ; this is indeed obvious, for the angles DEC, DCB are together half of the angles ABC, ACB, which themselves are tc^ether less than two right angles, and therefore the two bisectors of the angles B, C must meet, by Post. 5. It follows from the proof of this proposition that, if the bisectors of two angles B, C q( b. triangle meet in D, the line joining D ia A also bisects the third angle A, or the bisectors of the three angles. of a* triangle meet in a point It will be observed that Euclid uses the indirect form of proof when showing that the circle touches the three sides of the triangle. Simson proves it directly, and points out that Euclid does the same in 111. 17, 33 and 37, whereas in iv. 8 and 13 as well as here he uses the indirect form. The difference is unimportant, being one of fonn and not of substance; the indirect proof refers back to in. 16, whereas the direct refers back to the Porism to that proposition. We may state this problem in the moie general form : Te describe a circle touching three given straight lines which do net all meet in one point, and of which not mere than two are parallel. In the case (i) where two of the straight lines are parallel and the third cuts them, two pairs of interior angles are formed, one on each side of the third straight line. If we bisect each of the interior angles on one side, the bisectors will meet in a point, and this point will be the centre of a circle which can be drawn touching each of the three straight lines, its radius being the perpendicular from the point on any one of the three. Since the alternate angles are equal, two equal circles can be drawn in this manner satisfying the given condition. In the case (2) where the three straight lines form a triangle, suppose each straight line produced indefinitely. Then each straight line will make two pairs of interior angles with the other two, one pair forming two angles of the triangle, and the other pair being their supplements. By bisecting each angle of either pair we obtain, in the manner of the proposition, two circles satisfying the conditions, one of them tieing the inscribed circle of the triangSe and the other being a circle eseriied to it, i.e. touching one side and the other IV. 4] PROPOSITION 4 two sides ptvduftd. Next, taking the pairs of interior angles formed by a second side with the other two produced indefinitely, we get two circles satisfying the conditions, one of which is the same inscribed circle that we had before, while the other is a second escribed circle. Similarly with the third side. Hence we have the inscribed circle, and three escribed circles (one opposite each angle of the triangle), i.e. four circles in all, satisfying the conditions of the probiem. it ntay perhaps not be inappropriate to give at this point Heron's elegant proof of the formula for the area of a triangle in terms of the sides, which we usually write thus : A=Js{f-a){s-b){s-c), although it requires the theory of proportions and uses some ungeometrical expressions, e.g. the product of two areas and the "side " of such a product, where of course the areas are so many square units of length. The proof is given in the Metrica, i. 8, and in the Dioptra, 30 (Heron, Vol. iii., Teubner, 190J, pp. *o— i4 and pp. j8o — 4, or Heron, ed. Hultsch, pp. 235 — 7). Suppose the side-s of the triangle ABC\o be given in length. Inscribe the circle DEF, and let G be its centre. If ■! Join AG, BG, CG, DG, EG, FG. Then EC. EG =2. A BGC, CA. FG=i.£:.ACG, AB.DG=2.£.ABG. Therefore, by addition, p.EG^i.CiABC, where/ is the perimeter. Produce CB to H, so that BH^ AD. Then, since AD = AF, DB = BE, FC = CE, CH= y. Hence CH. EG=t, ABC. as BOOK IV [iv. 4, 5 But CH .EG is the "side" of the product CH^ . EC, that b JCH^.EG^; therefore {i^ABC)^=Cir.EG\ Draw GL at right angles to CG, and BL at right angles to CB, meeting at L. Join CL. Then, since each of the angles CGL, CBL is right, CGBL is a quadri- lateral in a circle. Therefore the angles CGB, CLB are equal to two right angles. Now the angles CGB, AGD are equal to two right angles, since AG, BG, CG bisect the angles at G, and the angles CGB, AGD are equal to the angles AGC, DGB, while the sum of all four is equal to four right angles. Therefore the angles j4 CZJ, CZZf are equal. So are the right angles ADG, CBL. Therefore the triangles AGD, CLB are similar. Hence BC: BL = AD-.DG = BH: EG, and, alternately, CB : BH = BL : EG = BK: KE, whence, tomponende, CH: HB = BE : EK. It follows that CH-" : CH . HB ^ BE . EC.CE . EK '^ BE. EC: EG* Therefore (A ABC)^ = CH' . EG'= CH. HB . CE . EB ^\P{\p-BC){\p-AB){\p-AO. Proposition 5. About a ^ven triangle to circumscribe a circle. Let ABC be the given triangle ; thus it is required to circumscribe a circle about the given triangle ABC. Let the straight lines AB, AC h& bisected at the points D, E [i. 10], and from the points D, E let DF, EF be drawn at right angles to AB, AC ; they will then meet within the triangle ABC^ or on the straight line BC, or outside BC. IV. 5j PROPOSITIONS 4, s 89 First let them meet within at /% and let FB, FC, FA be joined. Then, since AD is equal to DB, ■ . f and DF is common and at right angles, therefore the base AFis equal to the base FB. [!■ 4] Similarly we can prove that CF is also equal to ^^; ' so that FB is also equal to FC ; ^ therefore the three straight Hnes FA, FB, FC are equal to one another, Therefore the circle described with centre F and distance one of the straight lines FA, FB, FC will pass also through the remaining points, and the circle will have been circum- scribed about the triangle ABC. Let it be circumscribed, as ABC. Next, let DF, EF meet on the straight line BC at F, as is the case in the second figure ; and let AF be joined. Then, similarly, we shall prove that the point F is the centre of the circle circumscribed about the triangle ABC. Again, let DF, EF meet outside the triangle ABC at F, as is the case in the third figure, and let AF, EF, CF be joined. Then again, since AD is equal to DB, and DF'is common and at right angles, therefore the base AF is equal to the base BF. [i. 4] Similarly we can prove that C/^ is also equal to v^/^; so that BF is also equal to FC ; therefore the circle described with centre F and distance one of the straight lines FA, FB, FC will pass also through the remaining points, and will have been circumscribed about the triangle ABC. Therefore about the given triangle a circle has been circumscribed, Q. E. F. And it is manifest that, when the centre of the circle falls within the triangle, the angle BAC, being in a segment greater than the semicircle, is less than a right angle ; 90 BOOK IV [iv. s when the centre falls on the straight line BC, the angle BAC, being in a semicircle, is right ; and when the centre of the circle falls outside the triangle, the angle BAC, being in a segment less than the semicircle, is greater than a right angle, [m. 31] Sim son points out that Euclid does not prove that DF, EFmW meet, and he inserts in the text the following argument to supply the omission. "^ DF, ^/i" produced meet one another. For, if they do not meet, they are parallel, wherefore AB, AC, which are at right angles to them, are parallel [or, he should have added, in a straight line] : which is absurd." This assumes, of course, that straight lines which are at right angles to two parallels are themselves parallel ; but this is an obvious deduction from J. 28. On the assumption that DF, EF will meet Todhunter has this note : " It has been proposed to show this in the following way; join DE\ then the angles EDFi.nA. DEFax^ together less than the angles ADFand AEF, that is, they are together less than two right angles ; and therefore DF and Efi will meet, by Axiom 1 2 [Post. 5]. This assumes that ADE and AED are acute angles ; it may, however, be easily shown that DE is parallel to BC, so that the triangle ADE is equiangular to the triangle ABC; and we must therefore select the two sides AB and A C such that ABC and ACB may he acute angles." This is, however, unsatisfactory, Euchd makes no such selection in tti. 9 and III. 10, where the same assumption is tacitly made; and it is unnecessary, because it is easy to prove that the straight lines DF, EF meet in all cases, by considering the different possibilities separately and drawing a separate figure for each case. Sim son thinks that Euclid's demonstration had been spoiled by some unskilful hand both because of the omission to prove that the perpendicular bisectors meet, and because "without any reason he divides the proposition into three cases, whereas one and the same construction and demonstmtion serves for them all, as Cam pan us has observed," However, up to the usual words awiji (S(i jrmtjcrat there seems to be no doubt about the text. Heiberg suggests that Euclid gave separately the case where /"falls on BC because, in that case, only -^Z" needs to be drawn and not BF, CF a.s well. The addition, though given in Simson and the text-books as a "corollary," has no heading jropio-/t« in the best mss. ; it is an explanation like that which is contained in the penultimate paragraph of iii. 25. The Greek text has a further addition, which is rejected by Heiiwrg as not genuine, "So that, further, when the given angle happens to be less than a right angle, DF, EF will fall within the triangle, when it is right, on BC, and, when it is greater than a right angle, outside BC. (being) what it was required to do." Simson had already observed that the text here is vitiated " where mention is made of a given angle, though there neither is, nor can he, any- thing in the proposition relating to a given angle." IV. s, 6] PROPOSITIONS s, 6 ( Proposition 6. In a given circle to inscribe a square. • ' Let A BCD be the given circle ; thus ii is required to inscribe a square in the circle A BCD. Let two diameters AC, BD of the circle ABCD be drawn at right angles to one another, and let AB, BC, CD, DA be joined. Then, since BE is equal to ED, for E is the centre, and EA is common and at right angles, therefore the base AB is equal to the base AD. [[. 4] For the same reason each of the straight lines BC, CD is also equal to each of the straight lines AB, AD ; therefore the quadrilateral ABCD is equilateral, I say next that it is also right-angled. For, since the straight line BD is a diameter of the circle ABCD, • therefore BAD is a semicircle ; therefore the angle BAD is right. [ni. 31] For the same reason each of the angles ABC, BCD, CD A is also right ; therefore the quadrilateral ABCD is right-angled. But it was also proved equilateral ; therefore it Is a square ; [1. Def. 2»] and it has been inscribed in the circle ABCD. Therefore in the given circle the square ABCD has been inscribed. ■ " Q. E. F. Euclid here proceeds to consider problems conespondtng to those in Props. 2 — s with reference to figures of four or more sides, but with the difference that, whereas he dealt with triangles of any fomn, he confines him^lf henceforth to regular figures. It happened to be as easy to divide a circle into thrct parts which are in the ratio of the angles, or of the supplements of the angles, of a triangle as into three c^wn/ parts. But, when it is required to inscribe in a circle a figure equiangular to a given quadrUattral, this can only be 9* BOOK IV [iv. 6, 7 donu provided (hat the quadritateral has either pair of opposite angles equal to two right angles. Moreover, in this case, tlie problem may be solved in the same way as that of iv, z, i.e. by simply inscribing; a triangle equiangular to one of the triangles into which the quadrilateral is divided by either diagonal, and then drawing on the side corresponding to the diagonal as base another triangle equiangular Co the other triangle contained in the quadrilateral. But this is not the on/y solution ; there are an infinite number of other solutions in which the inscribed quadrilateral will, unlike that found by this particular method, not he of the same /arm as the given quadri- lateral For suppose A BCD to be the quadrilateral 11/ ^•'^ l^lry inscribed in the circle by the method of iv. 2. Take '■^'^ '"^ any point ff on AB, join AB", and then make the angle DAD (measured towards AC) equal to the angle BAff. Join ffC, CU. Then AECD is also tquiangular to the given quadrilateral, but not of the same form. Hence the problem is indeterminate in the case of the general quadrilateral. It is equally so if the given quadrilateral is a rectangle ; and it is determinate only when the given quadrilateral is a square. I \ Proposition 7. Aboui a given circle to circumscribe a square. Let ABCD be the given circle ; thus it is required to circumscribe a square about the circle ABCD. Let two diameters AC, BD of the circle ABCD be drawn at right angles to one another, and through the points A, B, C, D let FG, GH, HK, KF be drawn touching the circle ABCD. [III. i6,_Por.] Then, since FG touches the circk ABCD, and EA has been joined from the centre E to the point of contact at A, therefore the angles at A are right. [11 1. 18] For the same reason the angles at the points B, C, D are also right. Now, since the angle AEB is right, and the angle EBG is also right, therefore GH is parallel to AC. [i. 18^ IV. 7] PROPOSITIONS 6, 7 93 For the same reason AC is also parallel to FK, so that G// is also parallel to FK. . . ['• 3°] Similarly we can prove that each of the straight lines GF, HK is parallel to BED, Therefore GK, GC, AK, FB, BK are parallelograms ; therefore GF is equal to HK, and GH to FK. [i. 34] And, since AC is equal to BD, and AC is also equal to each of the straight lines G//, FA', while BO is equal to each of the straight lines GF, HK, [' 34] therefore the quadrilateral FGHK is equilateral. I say next that it is also right-angled. For, since GBEA is a parallelogram, and the angle AEB is right, therefore the angle AGB is also right. ^ [i. 34] Similarly we can prove that the angles at H, K, F are also right. Therefore FGHK is right-angled. But it was also proved equilateral ; therefore it is a square ; and it has been circumscribed about the circle ABCD. Therefore about the given circle a square has been circumscribed. . ■'^-" Q. E. F. I[ is just as easy to describe about a given circle a polygon equiangular to any given polygon as it is to describe a square about a given circle. We have only to use the method of iv. 3, i.e. to take any radius of the circle, to measure round the centre successive angles in one and the same direction equal to the supplements of the successive angles of the given polygon and, lastly, to draw tangents to the circle at the extremities of the several radii so detemiined ; but again the polygon would in general not be of the same form as the given one ; it would only be so if the given polygon happened to be such that a circle couid be inscribed in it. To take the case of a quadrilateral only : it is easy to prove that, if a quadrilateral be described about a circle, the sum of one pair of opjxtsite sides must be equal to the sum of the other pair. It may be proved, conversely, tliat, if a quadrilateral has the sums of the pairs of opposite sides equal, a circle can be inscribed in it. If then a given quadrilateial has the sums of the pairs of opposite sides equal, a quadrilateral can be described about any given circle not only equiangular with it but having the iaxa^form or, in the words of Book vi., similar to it. BOOK IV [iv. S [•• 34] Proposition 8. In a given square to inscribe a circle. Let ABCD be the given square ; thus it is required to inscribe a circle in the given square ABCD. ^ Let the straight lines AD^ AB be bisected at the points E, F respectively [»• to]. through E let EH be drawn parallel to either AB or CD, and through F let FK be drawn parallel to either AD or BC; [1.31] therefore each of the figures AK, KB, AH, HD, AG, GC, BG, GD is a parallelogram, and their opposite sides are evidently equal. Now, since AD is equal to AB, and ^^ is half of AD, and AF half of AB, therefore AE is equal to AF, so that the opposite sides are also equal ; • ' - -■ therefore FG is equal to GE. Similarly we can prove that each of the straight lines GH, GK is equal to each of the straight lines FG, GE ; therefore the four straight lines GE, GF, GH, GK are equal to one another. Therefore the circle described with centre G and distance one of the straight lines GE, GF, . GH, GK will pass also through the remaining points. And it will touch the straight lines AB, BC, CD, DA, because the angles at E, F, H, K are right. For, if the circle cuts AB, BC, CD, DA, the straight line drawn at right angles to the diameter of the circle from its extremity will fall within the circle : which was proved absurd ; [iii, ifi) therefore the circle described with centre G and distance one of the straight lines GE, GF, GH, GK will not cut the straight lines AB, BC, CD, DA. Therefore it will touch them, and will have been inscribed in the square ABCD. Therefore in the given square a circle has been inscribed. IV. 8, 9] PROPOSITIONS 8, 9 9$ As was remarked in the Iftst note, a circle can be inscribed in any ^uadrilatiral -vihKh has the sum of one pair ofoppc^ite sides equal to the sum of the other pair. In particular, il follows that a circle can be inscribed in a tfuare or a rhsmbus^ but not in a rectangle or a rhomboid. ' * Proposition 9, About a given square to circumscribe a circle. Let A BCD be the given square ; thus it is required to ctrcu inscribe a circle about the square A BCD. For let AC, BD be joined, and let them cut one another at E, Then, since DA is equal to AB^ and AC is common, therefore the two sides DA, AC are equal to the two sides BA, AC; and the base DC is equal to the base BC ; therefore the angle DAC is equal to the angle BAC. [i. 8] Therefore the angle DAB is bisected by AC. Similarly we can prove that each of the angles ABC, BCD, CDA is bisected by the straight lines AC, DB. Now, since the angle DAB is equal to the angle ABC, and the angle BAB is half the angle DAB, and the angle BBA half the angle ABC, 'V therefore the angle BAB is also equal to the angle £BA ; so that the side £A is also equal to BB. [i. 6] Similarly we can prove that each of the straight lines BA, BB is equal to each of the straight lines EC, ED. Therefore the four straight lines EA, BB, BC, BD are equal to one another. Therefore the circle described with centre E and distance one of the straight lines EA, EB, EC, ED will pass also through the remaining points ; and it will have been circumscribed about the square ABCD. Let it be circumscribed, as ABCD. Therefore about the given square a circle has been circumscribed. 96 BOOK IV [iv. lo PROPOSITtON 10. '■'- . ■ .• Ta cottslrticl an isosceles triangle Itaving each of the angles at the base double of the remaining one. Let any straight line AB be set out, and let it be cut at the point C so that the rectangle contained by AB, BC is equal to the square on CA\ [•'■"] with centre A and distance AB let the circle BDE be described, and let there be fitted in the circle BDE the straight line BD equal to the straight line AC which is not greater than the diameter of the circle BDE, [iv. ,] Let AD, DC be joined, and let the circle ACD be circumscribed about the triangle A CD. ['V. S) Then, since the rectangle AB, BC is equal to the square on AC, and AC is equal to BD, therefore the rectangle AB, BC is equal to the square on BD. And, since a point B has been taken outside the circle ACD, and from B the two straight lines BA, BD have fallen on the circle ACD, and one of them cuts it, while the other falls on it, and the rectangle AB, BC is equal to the square on BD, therefore BD touches the circle ACD. [in. 37] Since, then, BD touches it, and DC is drawn across from the point of contact at D, therefore the angle BDC is equal to the angle DAC in the alternate segment of the circle. [in. 3*] Since, then, the angle BDC is equal to the angle DAC, let the angle CDA be added to each ; therefore the whole angle BDA is equal to the two angles CDA, DAC. IV, to] PROPOSITION ro 97 But the exterior angle BCD is equal to the angles CD A, DAC; [1.3a] therefore the angle BDA is also equal to the angle BCD. But the angle BDA is equal to the angle CBD. since the side AD is also equal to AB ; [i. s] so that the angle DBA is also equal to the angle BCD. Therefore the three angles BDA, DBA, BCD are equal to one another. And, since the angle DBC is equal to the angle BCD, the side BD is also equal to the side DC. [i. 6] But BD is by hypothesis equal to CA ; therefore CA is also equal to CD, so that the angle CD A is also equal to the angle DA C ; [i-S] therefore the angles CD A, DAC are double of the angle DA C. But the angle BCD is equal to the angles CD A, DAC; therefore the angle BCD is also double of the angle CAD. But the angle BCD is equal to each of the angles BDA, DBA , therefore each of the angles BDA, DBA is also double of the angle DAB. Therefore the isosceles triangle ABD has been constructed having each of the angles at the base DB double of the remaining one. Q. E. F. There is every reason to conclude that the connexion of the triangle constructed in this proposition with the regular pentagon, and the construction of the triangle itself, were the discovery of the Pythagoreans. In the first place the Scholium iv. No. a {Heiberg, Vol. v. p. 273) says " this Book is the discovery of the Pythagoreans." Secondly, the summary in Proclus (p. fi$, to) says that Pythagoras discovered "the construction of the cosmic figures," by which rnust be understood the five regular solids. Thirdly, lamblichus {yn. Pyth, c. 18, s. 38) quotes a story of Hippasus, " that he was one of the Pytha- goreans but, owing to his being the first to publish and write down (the con- struction of) the sphere arising from the twelve pentagons {rrpi in tuv hi&txa ir<kraywfijf), perished by shipwreck for his inipiety, having got credit for the discovery all the same, whereas everything belonged to HIM {intivcmrov dkSpot), for it is thus that they refer to Pythagoras, and they do not call him by his name." Cantor has (i,, pp. 176 sqq.) collected notices which help us to form an idea how the discovery of the Euclidean construction for a regular pentagon may have been arrived at by the Pythagoreans. Plato puts into th". mouth of Timaeus a description of the formation from \ BOOK IV [iv. 10 right-angled triangles of the figures which are the faces o( the first four regular solids. The face of the cube is the S(]uare which is formed from isosceles right-angled triangles by placing four of these triangles contiguously so that the four right angles are in contact at the centre. The equilateral triangle, however, which is the form of the faces of the tetrahedron, the octahedron and the icosahedron, cannot be constructed from isosceles right-angled triangles, but is constructed from a particular scalene right-angled triangle which Timaeus (54 a, h) regards as the most b<:autiful of all scalene right-angled triangles, namely that in which the square on one of the sides about the right angle is three times the square on the other. This is, of course, the triangle forming half of an equilateral triangle bisected by the perpendicular from one angular point on the opposite side. The Platonic Timaeus does not construct his equilateral triangle from two such triangles but from six, by placing th« latter contiguously round a point so that the hypotenuses and the smaller of the sides about the right angles respectively adjoin, and all of them meet at the common centre, as shown in the figure {T/macHs, 54 d, k.). The probability that this exposition was Pythagorean is confirmed by the independent testimony of Proclus (pp. 304—5), who attributes to the Pythagoreans the theorem that six equilateral triangles, or three hexagons, or four squar^ placed contiguously with one angular point of each at a common point, will just fill up the four right angles round that point, and that no other regular polygons in any numbers have this property. How then would it be proposed to split up into triangles, or to make up out of triangles, the face of the remaining solid, the dodecahedron ? It would easily be seen that the pentagon could not be constructed by means of tlie two right-angled triangles which were used for constructing the square and the equilateral triangle respectively. But attempts would naturally be made to split up the pentagon into elementary triangles, and traces of such attempts are actually forthcoming. Plutarch has in two passages spoken of the division of the faces of the dodecahedron into triangles, remarking in one place (Quaes/. Platon. v. i) that each of the twelve faces is made up of 30 elemen- tary scalene triangles, so that, taken together, they give 360 such triangles, and in another {Dt deftctu oramlorum, c. 33) that the elementary triangle of the dodecahedron must be different from that of the tetrahedron, octahedron and icosahedron. Another writer of the and cent,, Alcinous, has, in his introduction to the study of Plato {De doctrina Platonis, c. 11), spoken similarly of the 360 elements which are produced when every one of the pentagons is divided into 5 isosceles triangles, and each of the latter into 6 scalene triangles. Now, if we proceed to draw lines in a pentagon sejmrating it into this number of small triangles as shown in the above figure, the figure IV. lo] PROPOSITION lo 99 which stands out most prominently in the mass of lines is the "star-pentagon," as drawn separately, which then (if the consecutive comers be joined) suggests the drawing, as part of a pentagon, of a triangle of a definite character. Now we are expressly told by Lucian and the scholiast to the Clouds of Aristophanes (see Bretschneider, pp. 85 — 86) that the triple interwoven triangle, the penta- gram (to TpurXflEf Tpiycuvov, TO Zi oAAijAaic, to TroTaypttfiftov), was used by the Pythagoreans as a symbol of recognition between the members of the same school {av^ohjjf ipos Tous ofioSo'fou! ixf™"'°)i *"d '"'** called by them Health. There seems lo be therefore no room for doubt that the construction of a pentagon by means of an isosceles triangle having each of its base angles double of the vertical angle was due to the Pythagoreans. The construction of this triangle depends upon ii. 1 1, or the problem of dividing a straight tine so that the rectangle contained by the whole and one of the parts is equal to the square on the other part. This problem of course appears again in Eucl. vi. 30 as the problem of cutting a given straight line in exirtme and mean ratio, i.e. the problem of the goldtn section, which is nc- doubt " the section " referred to in the passage of the summary given by Proclus (p. 67, 6) which says that Eudoxus "greatly added to the number of the theorems which Plato originated regarding the section." This idea that Plato began the study of the " golden section " as a subject in itself is not in the least inconsistent with the supposition that the problem of Eucl ii. 1 1 was solved by the Pythagoreans. The very fact that Euclid places it among other propositions which are clearly Pythagorean in origin is significan|, as is also the fact that its solution is effected by " applying to a straight line a rectangle equal to a given square and exceeding by a square," while Proclus says plainly (p. 419, 15) that, according to Eudemus, "the application of areas, their txcuding and their falling short, are ancient and discoveries of the Muse of the Pythagoreans." We may suppose the construction of iv. to to have been arrived at by analysis somewhat as follows (Todhunter's Euclid, p. 325). Suppose the problem solved, i.e. let ABD be an isosceles triangle having each of its base angles double of the vertical angle. Bisect the angle ADB by the straight line DC meeting AB in C. [1, 9] Therefore the angle BDC is equal to the angle BAD ; and the angle CDA is also equal to the angle BAD, so that DC is equal to CA, Again, since, in the triangles BCD, BDA, the angle BDC is equal to the angle BAD, and the angle B is common, the third angle BCD is equal to the third angle BDA, and therefore to the angle DEC. Therefore Z'C is equal to DB. Now, if a circle be described about the triangle ACD [iv. 5}, since the angle BDC is equal to the angle in the segment CAD, BD must touch the circle [by the converse of [ii. 32 easily proved from it by riduttio ad aiiurdum\ Hence [11 [. 36] the square oa BD and therefore the square on CD, or AC, is equal to the rectangle AS, BC. Thus the problem is reduced to that of cutting AB at C so that the rectangle AB, BC is equal to the square on AC, [ii, 11] t99 ' BOOK IV [iv. lo, It When this is done, we have only to draw a circle with centre A and radius A£ and place in it a chord SU equal in length to AC. [iv. i] Since each of the angles ABD, ADB is double of the angle BAD, the latter is equal to one-fifth of the sum of all three, i.e. is one-fifth of two right angles, or two-fifths of a right angle, and each of the base angles is four-fifths of a right angle. If we bisect the angle BAD, we obtain an angle equal to one-fifth of a right angle, so that the proposition enables us to divide a right angle into five equal parts. It will be observed that BD is the side of a regular dtcagen inscribed in the larger circle. Proclus, as retnarked above (Vol. i, p. 130), gives iv. 10 as an instance in which two of the six formal divisions of a proposition, the idting-out and the "definition" are left out, and explains that they are unnecessary because there is no datum in the enunciation. This is however no more than formally true, because Euclid does begin bis proposition by tetling out " any straight line AB" and he constructs an isosceles triangle having AB for one of its equal sides, i.e. he does practically imply a datum in the enunciation, and a corresponding setting-out and "definition '' in the proposition itself. Proposition 11. In a given circle to inscribe an equilateral and ^uiangular pentagon. Let ABCDE be the given circle ; thus it is required to inscribe in the circle ABCDE an equi- lateral and equiangular pentagon. Let the isosceles triangle FGH be set out having each of the angles at G, H double of the angle at F\ [iv. 10] let there be inscribed in the circle ABCDE the triangle ACD equi- angular with the triangle FGH, so that the angle CAD is equal to the angle at F and the angles at G, ^respectively equal to the angles ACD, CD A ; [iv. a] therefore each of the angles ACD, CD A is also double of the angle CAD. Now let the angles ACD, CD A be bisected respectively by the straight lines CE, DB [1. 9], and let AB, BC, DE, EA be joined. Then, since each of the angles ACD, CD A is double of the angle CAD, and they have been bisected by the straight lines CE, DB, IV. ii] PROPOSmONS 10, II i«i therefore the five angles DAC, ACE, BCD, CDS, BDA are equal to one another. But equal angles stand on equal circumferences ; [iii. a6] therefore the five circumferences AB, BC, CD, DE, EA are equal to one another. But equal circumferences are subtended by equal straight lines ; [m. 19] therefore the five straight lines AB, BC, CD, DE, EA are equal to one another ; therefore the pentagon ABCDE is equilateral. I say next that it is also equiangular. For, since the circumference AB is equal to the circum- ference DE, let BCD be added to each ; therefore the whole circumference ABCD is equal to the whole circumference EDCB. And the angle A ED stands on the circumference ABCD, and the angle BAE on the circumference EDCB ; therefore the angle BAE is also equal to the angle AED. [iiL a;] For the same reason * each of the angles ABC, BCD, CDE is also equal to each of the angles BAE, AED ; therefore the pentagon ABCDE is equiangular. But it was also proved equilateral ; therefore in the given circle an equilateral and equi- angular pentagon has been inscribed. Q. E. F. De Morgan remarks that " the method of iv. 11 is not %a natuial as malting a direct use of the angle obtained in the last." On the other hand, if we look at the figure and notice that it shows the whole of the pmtagrafo- star except one line {that connecting B and E), I think we shall conclude that the method is nearer to that used by the Pythagoreans, and therefore of much more historical interest. Another method would of course be to use iv. 10 to describe a decagtnt in the circle, and then to join any vertex to the next alternate one, the tatter to the next alternate one, and so on. K». BOOK IV [iV. II, I » Mr H. M. Taylor gives "a complete geometrical construction for in- scribing a regular decagon or pentagon in a given circle," as follows. " Find O the centre. Draw two diameters AOC, BOD at right g angles to one another. Bisect OD in £. Draw A£ and cut off E£ equal to 0£. Place round the circle ten chords equal to AF. These chords will be the sides of a regular decagon. Draw the chords joining Hve alternate vertices of the decagon ; they will be the sides of a regular pentagon." The construction is of course only a com- bination of those in ]i. ii and iv. i ; and the proof would have to follow that in iv. lo. Proposition 12, About a given circle to circumscribe an equilateral and equiangular pentagon. Let y4 .5 CZ?.£' be the given circle ; >- thus it is required to circumscribe an equilateral and equi- angular pentagon about the circle ABCDE. Let A, B, C, D, E be conceived to be the angular points of the inscribed pentagon, so that the circumferences AB, EC, CD, DE, EA are equal ; through A, B, C, D. E let G//, HK, KL, LM, MG be drawn touching the circle ; [in. 16, Por.] let the centre F of the circle ABCDE be taken [m. 1], and let FB, FK, FC, FL, FD be joined. Then,since the straight line KL touches the circle ABCDE at C, and FC has been joined from the centre F to the point of contact at C, therefore FC is perpendicular to KL ; , [in. 18] therefore each of the angles at C is right. • ■ For the same reason the angles at the points B, D are also right IV. laj PROPOSITIONS ii, la 103 And, since the angle FCK is right, therefore the square on FK is equal to the squares on FC, CK. For the same reason [1. 47] the square on FK is also equal to the squares on FB, BK ; so that the squares on FC, CK are equal to the squares on FB, BK, of which the square on FC is equal to the square on FB ; therefore the square on CK which remains is equal to the square on BK. Therefore BK is equal to CK. And, since ^5 is equal to T^C • • and FK common, the two sides BF, FK are equal to the two sides CF, FK \ and the base BK equal to the base CK ; therefore the angle BFK is equal to the angle KFC, [i. 8] and the angle BKF to the angle FKC. Therefore the angle BFC is double of the angle KFC, and the angle BKC of the angle FKC. For the same reason the angle CFD is also double of the angle CFL, and the angle DLC of the angle FL C. Now, since the circumference BC is equal to CD, the angle BFC is also equal to the angle CFD. [in. 17] And the angle BFC is double of the angle KFC, and the angle DFC of the angle LFC ; therefore the angle KFC is also equal to the angle LFC. But the angle FCK is also equal to the angle FCL ; therefore FKC, FLC are two triangles having two angles equal to two angles and one side equal to one side, namely FC which is common to them ; therefore they will also have the remaining sides equal to the remaining sides, and the remaining angle to the remaining angle ; [i- *6) therefore the straight line KC is equal to CL, and the angle FKC to the angle FLC, And, since KC is equal tO CZ, , therefore KL is double of KC , . 104 BOOK IV [iv. ti, 13 For the same reason it can be proved that I/J^ is also double of S/C. 1 And ^A' is equal to ^C; ■ therefore I/K is also equal to KL. Similarly each of the straight lines //G, GM, ML can also be proved equal to each of the straight lines //K, KL ; therefore the pentagon GHKLM is equilateral. . - I say next that it is also equiangular. For, since the angle FKC is equal to the angle FLC, and the angle HKL was proved double of the angle FKC, and the angle KLM double of the angle FLC, therefore the angle HKL is also equal to the angle KLM. Similarly each of the angles KHG, HGM, GML can also be proved equal to each of the angles HKL, KLM; therefore the five angles GHK, HKL, KLM, LMG, MGH are equal to one another. Therefore the pentagon GHKLM is equiangular. And it was also proved equilateral ; and it has been circumscribed about the circle ABCDE. Q. E. F. De Morgan remarks that iv. 12, 13, 14 supply the pkce of the following : Having given a regular polygon of any number of sides inscribed in a circle, lo describe the same about ike circle; and, having given the polygon, lo inscribe and circumscribe a circle. For the method can be applied generally, as indeed Euclid practically says in the Porism to iv. 15 about the regular hexagon and in the remark appended to iv. 16 about the regular fifteen-angled figure. The conclusion of this proposition, " therefore about the given circle an equilateral and equiangular pentagon bas been circumscnbed," is omitted in the Mss. Proposition 13. In a given pentagon, wkick is equilateral and equiangular, to inscribe a circle. Let ABCDE be the given equilateral and equiangular pentagon ; thus it is required to inscribe a circle in the pentagon ABCDE. For let the angles BCD, CDE be bisected by the straight lines CF, Z?J^ respectively ; and from the point F, at IV. 13] PROPOSITIONS 12, 13 105 ■'IJ-/J which the straight lines CF, DF meet one another, let the straight lines FB, FA, FE be joined. 1 • • Then, since BC is equal to CD, and CV^ common, the two sides BC, CF are equal to the two sides DC, CF\ and the angle BCF is equal to the angle DCF; therefore the base BF is equal to the base DF, and the triangle BCF is equal to the triangle DCF, and the remaining angles will be equal to the remaining angles, namely those which tne equal sides subtend. [1. 4] Therefore the angle CBF is equal to the angle CDF. And, since the angle CDE is double of the angle CDF, and the angle CDE is equal to the angle ABC, while the angle CDF is equal to the angle CBF\ therefore the angle CBA is also double of the angle CBF) therefore the angle ABF is equal to the angle FBC ; therefore the angle ABC has been bisected by the straight line BF. Similarly it can be proved that the angles BAE, AED have also been bisected by the straight lines FA, FE respectively. Now let EG, FH, FK, FL, FMh& drawn from the point F perpendicular to the straight lines AB, BC, CD, DE, EA. Then, since the angle HCF is equal to the angle KCF, and the right angle FHC is also equal to the angle FKC, FHC, FKC are two triangles having two angles equal to two angles and one side equal to one side, namely EC which is common to them and subtends one of the equal angles ; therefore they will also have the remaining sides equal to the remaining sides ; [i. i6] therefore the perpendicular FH is equal to the perpendicular FK. Similarly it can be proved that each of the straight lines FL, FM, EG is also equal to each of the straight lines EH, FK ; i«6 BOOK IV [iv. 13, u therefore the five straight lines FG, FH, FK, FL, FM are equal to one another. Therefore the circle described with centre F and distance one of the straight lines FG, FH, FK, FL, FM will pass also through the remaining points ; and it will touch the straight lines AB, BC, CD, DE, EA, because the angles at the points G, N, K, L, M aire right. For, if it does not touch them, but cuts them, it will result that the straight line drawn at right angles to the diameter of the circle from its extremity falls within the circle : which was proved absurd. [ui. 16] Therefore the circle described with centre F and distance one of the straight lines FG, FN, FK, FL, FM will not cut the straight lines AB, BC, CD, DE, EA ; therefore it will touch them. Let it be described, as GHKLM. Therefore in the given pentagon, which is equilateral and equiangular, a circle has been inscribed, Q, E. F. Proposition 14. About a given pentagon, which ts equilateral and equi- angular, to circumscribe a circle. Let ABCDE be the given pentagon, which is equilateral and equiangular ; thus it is required to circumscribe a circle about the pentagon ABCDE. Let the angles BCD, CDE be bisected by the straight lines CF, OF respectively, and from the point F, at which the straight lines meet, let the straight lines FB, FA, FE be joined to the points B, A, E. Then in manner similar to the pre- ceding it can be proved that the angles CBA, BAE, AED have also been bisected by the straight lines FB, FA, FE respectively. • IV. 14, is] PROPOSITIONS 13— IS 107 Now, since the angle BCD is equal to the angle CDE, and the angle FCD is half of the angle BCD, and the angle CDF half of the angle CDE, therefore the angle FCD is also equal to the angle CDF, ■' ! so that the side FC is also equal to the side FD. [i- 6] Similarly it can be proved that each of the straight lines FB, FA, FE is also equal to each of the straight lines FC, FD ; therefore the five straight lines FA, FB, FC, FD, FE are equal to one another. Therefore the circle described with centre F and distance one of the straight lines FA, FB, FC, FD, FE will pass also through the remaining points, and will have been circumscribed. , , Let it be circumscribed, and let it be ABCDE. Therefore about the given pentagon, which is equilateral and equiangular, a circle has been circumscribed. Q. E, F. Proposition 15. In a given, circle to inscribe an equilateral and equiangular hexagon. Let ABCDEF be the given circle ; ,...::; thus it is required to inscribe an equilateral and equiangular hexagon in the circle ABCDEF. Let the diameter AD of the circle ABCDEF be drawn ; let the centre G of the circle be taken, and with centre D and distance DG let the circle EGCH be described ; let EG, CG be joined and carried through to the points B, F, and let AB, EC, CD, DE, EF, FA be joined. I say that the hexagon ABCDEF is equilateral and equiangular. For, since the point G is the centre of the circle ABCDEF, GE is equal to GD. ■■ v. •• v •»*S BOOK IV 1 " [iv, IS Again, since the point D is the centre of the circle GCH, DE is equal to DG, ^ ■':-' „ But GE was proved equal to GD ; therefore GE is also equal to ED ; therefore the triangle EGD is equilateral ; and therefore its three angles EGD, GDE, DEG are equal to one another, inasmuch as, in isosceles triangles, the angles at the base are equal to one another. [i. s] And the three angles of the triangle are equal to two right angles ; [i. 3^1 therefore the angle EGD is one-third of two right angles. Similarly, the angle DGC can also be proved to be one- third of two right angles. And, since the straight line CG standing on EB makes the adjacent angles EGC, CGB equal to two right angles, therefore the remaining angle CGB is also one-third of two right angles. Therefore the angles EGD, DGC, CGB are equal to one another ; so that the angles vertical to them, the angles SGA, AGF, FGE are equal, [i. 15] Therefore the six angles EGD, DGC, CGB, BGA, AGF, FGE are equal to one another. But equal angles stand on equal circumferences ; {m- a6] therefore the six circumferences AB, BC, CD, DE, EF, FA are equal to one another. And equal circumferences are subtended by equal straight lines ; [m. 29] therefore the six straight lines are equal to one another; therefore the hexagon ABCDEF is equilateral, I say next that it is also equiangular. For, since the circumference FA is equal to the circum- ference ED, let the circumference ABCD be added to each ; therefore the whole FA BCD is equal to the whole EDCBA ; IV, is] proposition is 109 and the angle FED stands on the circumference FA BCD, and the angle AFE on the circumference EDCBA ; therefore the angle AFE is equal to the angle DEF, [m. 27] Similarly it can be proved that the remaining angles of the hexagon ABCDEF are also severally equal to each of the angles AFE, FED ; therefore the hexagon ABCDEF is equiangular. But it was also proved equilateral ; and it has been inscribed in the circle ABCDEF. Therefore in the given circle an equilateral and equiangular hexagon has been inscribed. Q. E. F. PoRiSH. From this it is manifest that the side of the hexagon is equal to the radius of the circle. And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle an equilateral and equiangular hexagon in conformity with what was explained in the case of the pentagon. And further by means similar to those explained in the case of the pentagon we can both inscribe a circle in a given hexagon and circumscribe one about it, , Q. E. F. Hetberg, I think with good reason, considers the Porism to this proposition to be referred to in the instance which Proclus (p. 304, a) gives of a porism following a problem. As the text of Proclus stands, " the (poristn) found in the second Book (td ii ir ry Smrifnf ^ijSXi'u xtiiititor) is a porism to a problem " ; but this is not true of the only porism that we find in the second Book, namely the jwrism to it. 4. Hence Heibeig thinks that for rif StvfifMf fiifiXiif should be read 1^ £' fii^Kuf, i.e. the fourth Book. Moreover Proclus speaks of tAe porism in the particular Book, from which we gather that there was only arte porism in BooJt iv. as he knew it, and therefore that he did not regard as a porism the addition to iv. 5. Cf. note on that proposition. It appears that Theon substituted for the first words of the Porism to IV. 15 "And in like manner as in the case of the pentagon" (d/iouiit Si ToTs M. rol leivTaymrm} the simple word " and " or " also " (Wj, apparently thinking that the words had the same meaning as the similar words lower down. This is however not the case, the meaning being that " if, as in the case of the pentagon, we draw tangents, we can prove, also as was done in the case of the pentagon, that the figure so formed is a circumscribed r^ular hexagon." BOOK. IV Proposition i6. [tv. i6 In a given circle to inscribe a fifteen-angled figure which shall be both equilateral and equiangular. Let ABCD be the given circle ; thus it is required to inscribe in the circle ABCD a fifteen- angled figure which shall be both equilateral and equi- angular. In the circle ABCD let there be inscribed a side AC of the equilateral triangle inscribed in it, and a side AB of an equilateral pentagon ; therefore, of the equal seg- ments of which there are fifteen in the circle ABCD, there will be five in the cir- cumference ABC which is one-third of the circle, and there will be three in the cir- cumference AB which is one-fifth of the circle ; therefore in the remainder BC there will be two of the equal segments. Let BC be bisected at E ; [m. 30] therefore each of the circumferences BE, EC is a fifteenth of the circle ABCD, If therefore we join BE, EC and fit into the circle ABCD straight lines equal to them and in contiguity, a fifteen-angled figure which is both equilateral and equiangular will have been inscribed in it. ' ' ■ ' Q. E. F. And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle a fifteen-angled figure which is equilateral and equi- angular. And further, by proofs similar to those in the case of the pentagon, we can both inscribe a circle in the given fifteen- angled figure and circumscribe one about it. JV. .6] PROPOSITION tfi ui Here, as in ii[. lo, we have the term "circle" used by Euclid in its exceptional sense of the drcumjerena of a circle, instead of the "plane figurt contained by one hne" of i. l)ef. 15. Cf. the note on that definition (Vol. i. pp. 184—5}. Proclus {p. 269) refers to this proposition in illustratiotv of his statement that Euclid gave proofs of a number of propositions with an eye to their use in astronomy. " With regard to the last proposition in the fourth Book in which he inscribes the side of the fifteen-angled figure in a circle, for what object does anyone assert that he propounds it except for the reference of this problem to astronomy ? For, when we have inscribed the fifteen -angled figure in the circle through the poles, we have the distance from the poles both of the equator and the zodiac, since they are distant from one another by the side of the fifteen-angled figure," This agrees with what we know from other sources, namely that up to the time of Eratosthenes {circa 2iJ4 -204 B.C.) 24 was generally accepted as the correct measurement of the obliquity of the ecliptic. This measurement, and the construction of the fifteen-angled figure, were probably due to the Pythagoreans, though it would appear that the former was not known to Oenopides of Chios {fl. circa 460 B.C.), as we learn from Theon of Smyrna {pp. 198 — 9, ed, Hiller), who gives Dercy Hides as his authority, that Eudemus (H. circa 32a B.C.) stated in his dcrTp«A.<ry('<u that, while Oenopides discovered certain things, and Thales, Anaximander and Anaximenes others, it was the rest (01 AoHrm) who added other discoveries to these and, among them, that " the axes of the fixed stars and of the planets respectively are distant from one another by the side of a fifteen-angled figure." Eratosthen« evaluated the angle to J^rds of 180°, i.e. about 23' 51' 10", which measurement was apparently not improved upon in antiquity (cf. Ptolemy, Syataxii, ed. Heiberg, p. 68). Euclid has now shown how to describe regular polygons with 3, 4, 5, £ and 15 sides. Now, when any regular polygon is given, we can construct a regular polygon with twice the number of sides by first describing a circle about the given polygon and then bisecting all the smaller arcs subtended by the sides. Applying this process any number of times, we see that we can by Euclid's methods construct regular polygons with 3,1", 4-a*, 5,2", 15.2" sides, where « is zero or any positive integer. .1 1, , \ . f ' ' BOOK V. INTRODUCTORY NOTE. The anonymous author of a scholium to Book v. (Euclid, ed. Heiberg, Vol. V. p. 280), who is perhaps E^oclus, tells us that "some say" thb Book, containing the general theory of proportion which ts equally applicable to geometry, arithmetic, music, and all mathematical science, "is the discovery of Eudoxus, the teacher of Plata" Not that there had been no theory of proportion developed before his time j on the contrary, it is certain that the Pythagoreans had worked out such a theory with regard to numbtrs, by which must be understood commensurable and even whole numbers {a number being a " multitude made up of units," as defined in Eucl. vii). Thus we are told that the Pythagoreans distinguished three sorts of means, the arithmetic, the geometric and the harmonic mean, the geometric mean being called proportion (amXoyui) par exallenee; and further lamblichus speaks of the "most perfect proportion consisting of four terms and specially called harmonU" in other words, the proportion a + b xab , ■'■'■■ ■ ■ "'^'IT-y^' which was said to be a discovery of the Babylonians and to have been Rrst introduced into Greece by Pythagoras (lamblichus, Comm. en Ni&mtachas, p. ti8). Now the principle of similitude is one which is presupposed by all the arts of design from their very beginnings ; it was certainly known to the Egyptians, and it must certainly have been thoroughly familiar to Pythagoras and his school. This consideration, together with the evidence of the employment by him of the g^emetric proportion, makes it indubitable that the Pythagoreans used the theory of proportion, in the form in which it was known to them, i.e. as applicable to commensurables only, in their geometry. But the discovery, also by the Pythagoreans, of the incommensurable would of course be seen to render the proofs which depended on the theory of proportion as then understood inconclusive ; as Tannery observes (Xrr Giomftrie grecqui, p. 98), *' the discovery of incommensurability must have caused a veritable logical scandal in geometry and, in order to avoid it, they were obliged to restrict as far as possible the use of the principle of similitude, pending the discovery of a means of establishing it on the basis of a theory of proportion independent of commensurability." The glory 0/ the latter dis- covery belongs then most probably to Eudoxus. Certain it is that the com. plete theory was already familiar to Aristotle, as we shall see later. V. DEFF. I, a] INTRODUCTORY NOTE uj. It seems probable, as indicated by Tannery {lot. a'i.}, that the theory of proportions and the principle of similitude took, in the earliest Greek geometry, an earlier place than they do in Euclid, but that, in consequence of the discovery of the incommensurable, the treatment of the subject was fundamentally remodelled in the period between Pythagoras and Eudoxus, An indication of this is afforded by the clever device used in Euclid i. 44 for applying to a given straight line a parallelogram equal to a given triangle ; the equality of the "complements" in a parallelc^iam is there used for doing what is practically finding a fourth proportional to three given straight lines. Thus Euclid was no doubt following for the subject-matter of Books t. — iv. what had become the traditional method, and this is probably one of the reasons why proportions and similitude are postponed till as late as Books v., VI, It is a remarkable fact that the theory of proportions is twice treated in Euclid, in Book v. with reference to magnitudes in general, and in Book vci. with reference to the particular case of numbers. The latter exposition referring only to commensurable^ may be taken to represent fairly the theory of proportions at the stage which it had reached before the great extension of it made by Eudoxus. The differences between the definitions etc. in Books v. and VII. will appear as we go on ; but the question naturally arises, why did Euclid not save himself so much repetition and treat numbers merely as a particular case of magnitude, referring back to the corresponding more general propositions of Book v. instead of proving the same propositions over again for numbers? It could not have escaped him that numbers fall under the conception of magnitude. Aristotle had plainly indicated that magnitudes may be numbers when he observed {Anal. post. t. 7, 75 b 4) that you cannot adapt the arithmetical method of proof to the properties of magnitudes if the magnitudes are not numbers. Further Aristotle had remarked {Anal. post. 1. 5, 74 a 17) that the proposition that the terms of a proportion can be taken alternately was at one time proved •eparately for numbers, lines, solids and times, though it was possible to prove it for all by one demonstration ; but, because there was no common tuime comprehending them all, namely numbers, lengths, times and solids, and their character was different, they were taken separately. Now however, he adds, the proposition is proved generally. Yet Euclid says nothing to connect the two theories of proportion even when he comes in x. 5 to a proportion two terms of which are magnitudes and two are numbers (" Com mensurable magnitudes have to one another the ratio which a number has to a number"). The probable explanation of the phenomenon is that Euclid simply followed tradition and gave the two theones as he found them. This would square with the remark in Pappus (vii. p. 678) as to Euclid's fairness to others and his readiness to give them credit for their work. DEFINITIONS. 1. A magnitude is a part of a magnitude, the less of the greater, when it measures the greater. 2. The greater is a multiple of the less when it is measured by the less. 114 ■ ' ' BOOK V ' [v. DKFF, 3—13 3. A ratio is a sort of relation in respect of size between two magnitudes of the same kind. . , 4. Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another. 5. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimuhiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. 6. Let magnitudes which have the same ratio be called proportional. 7. When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth. 8. A proportion in three terms is the least possible. 9. When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second. 10. When four magnitudes are < continuously > propor- tional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on con- tinually, whatever be the proportion. 11. The term corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents. 12. Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent. 13. Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent. V. DEFF.] DEFINITIONS 115 14. Composition of a ratio means taking the ante- cedent together with the consequent as one in relation to the consequent by itself. 15. Separation of a ratio means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself. 16. Conversion of a ratio means taking the ante- cedent in relation to the excess by which the antecedent exceeds the consequent. r< 17, A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes ; Or, in other words, it means taking the extreme terms by virtue of the. removal of the intermediate terms. 18. A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes. Definition i. The word/ar/ (fii'pot) is here used in the restricted sense of a submtiitipU or an aliquot part as distinct from the more general sense in which it is used in the Common Notion (;) which says that "the whole is greater than the part." It is used in th^ same restricted sense in vii. Def, 3, which is the same definition as this with "number" (opifl/iot) substituted for "magnitude." VII. Def. 4, keeping up the restriction, says that, when a number does not measure another (>umoer, it is farfs (in the plural), not b part of it. Thus, I, a, or 3, is a part of 6, bat 4 is not a pari of 6 but parts. The same distinction between the restricted and the more general sense of the word part appears in Aristotle, Mdaph. 1023 b is: "In one sense a part is that into which quantity (to irocrov) can anyhow be divided ; for that which is taken away from quantity, guA quatitity, is always called a 'part' of it, as e.g. two is said to be in a sense a part of three. But in another sense a 'part' ill only what mtasura (ra Karo/wTpovtra) such quantities. Thus two b in one sense said to be a part of three, in the other not." .%>t6 BOOK V [v. DEFF. a, 3 1. r - Definition 2. *< ►/r* noXXa)rX(i(rtov Si to ^«i£oy roC lAttrnivaft orov narajitTp^Tai vn> raC IXaTTOfot. Definition 3. AvytK ivrl Suo /iicyfPur ofiOytv^if ^ Kara m^Xifcon^ra vota <r^w($. The best explanation of the definitions of ratio miA proportion that I have seen is that of De Moigan, which will be found in the articles under those titles in the Penny Cyclopaedia, Vol xix. (1841) ; and in the following notes I shall draw largely from these articles. Very valuable also aie the notes on the definitions of Book v. given by Hanlcel (fragment on Euclid published as an appendix to his work Zur GeschiehU der Mathimaiik in AUtrthum und Mittdalier, 1874). There has been controversy as to what is the proper translation of the word in)Xucar)}s in the definition, irxitrit 'ara n-i^XtxcTTifTii has generally been translated " relation in respect of quantify." Upon this De Morgan remarks that it makes nonsense of the definition ; "for magnitude has hardly a different meaning from quantity, and a relation of magnitudes with respect to quantity may give a clear idea to those who want a word to convey a notion of architecture with respect to building or of battles with respect to fighting, and to no others." The true interpretation De Morgan, following Wallis and Gregory, takes to be guantuplidty, referring to the number of times one magnitude is contained in the other. For, he says, we cannot describe magnitude in language without quantuplicitative reference to other magni- tude; hence he supposes that the definition simply conveys the fact that the mode of expressing quantity in terms of quantity is entirely based upon the notion of quantuphcity or that relation of which we take cognizance when we find how many times one is contained in the other. While all the rest of De Morgan's observations on the definition are admirable, it seems to me that on ttiis question of the proper translation of infAtKo'Tijt he is in error. He supports his view tnainly by reference (i) to the definition of a compounded ratio usually given as the 5th definition of Book vi., which speaks of the TiiKiKonfrti of two ratios being multiplied together, and (t) to the comments of Eutocius and a scholiast on this definition. Eutocius says namely (Archimedes, ©d. Heiberg, iii, p. wo) that "the term njKuainp is evidently used of the number from which the given ratio is called, as (among others) Nicomachus says in his first book on music and Heion in his commentary on the Introduction to Arithmetic." But it now appears certain that this definition is an interpolation ; it is never used, it is not found in Campanus, and Peyrard's MS. only has it in the margin. At the same time it is clear that, if the definition is admitted at all, any commentator would be obliged to explain it in the way that Eutocius does, whether the explanation was consistent with the proper meaning of mfXutorijv or not. Hence we must look elsewhere for the meaning of m^XiKot and nrXtitttnTt. If we do this, I think we shall find no case in which the words have the sense attributed to them by De Morgan. The teal meaning of irufXi'itos is how great. It is so used by Aristotle, e.g. in Eth, Me. V. to, 1134 b 11, where he speaks of a man's child being as it were a part of him so long as he is of a certain age (lutt av ^ m^XtKcn'), Ag»in Nicomachus, to whom Eutocius appeals, himself (i, 2, 5, p. Si ed. Hoche) distinguishes ttjjXikov as referring to magnitude, while h-octo's refers to multitude. So does lamblichus in his commentary on Nicomachus (p. 8, 3 — 5) ; besides which lamblichus distinguishes irq^dKov as the subject of geometry, being am- V. DEF. 3] NOTE ON DEFINITION 3 117 tinucus, and irocraf as the subject of arithmetic, being discrele, and speaks of a point being the origin of Tnjkinov as a unit is of iroo-oV, and so on. Similarly, Ptolemy (Syntaxis, ed, Heiberg, p. 31) speaks of the sise (injXijtdr)^) of the chords in a circle (jr<pi njt mjXntoTifros tw jv rifi kuhXj^ ti6nwv). Consequently I think we can only translate wii\tK6n)t in the definition as size. This corresponds to Hankel's translation of it as " GrOsse," though he uses this same word for a concrete " magnitude " as well ; size seems to me to give the proper distinction between injXiitontt and ii,iyt0o^, as size is the attribute, and a magnitude {in its ordinary mathematical sense) is the thing which possesses the attribute of siie. The view that " relation in respect of iize " is meant by the words in the text is also confirmed, I think, by a later remark of De Morgan himself, tiamely that a synonym for the word raiis may be found in the more in- telligible term relative magniiude. In fact axvm in the definition corresponds to relative and in/XotoTj)^ to magnitude. (By magnitude De Mo^an here means the attribute and not the thing possessing it.) Of the definition as a whole Simson and Hankel express the opinion that it is an interpolation. Hankel points to the fact that it is unnecessary and moreover so vague as to be of no practical use, while the very use of the expression na™ mjXtitorijTa seems to him suspicious, since the only other place in which the word wrjkiKoxTp occurs in Euclid is the 5th definition of Book VI., which is admittedly not genuine. Yet the definition of ratio appears in all the MSS., the only variation being that some add the words npm oAAifXo, "to one another," which are rejected by Heiberg as an interpolation of Theon ; and on the whole there seems to be no sufficient ground for regarding it as other than genuine. The true explanation of its presence would appear to be substantially that given by Barrow {Lectiones Cantabrig., London, 1684, Lect. Ill, of 1666), namely that Euclid bserted it for completeness' sake, mote for ornament than for use, intending to give the learner a general notion of ratio by means of a metaphysical, rather than a mathematical definition ; " for metaphysical it is and not, properly speaking, mathematical, since nothing depends on it or is deduced from it by mathematicians, nor, as I think, can anything be deduced." This is confirmed by the fact that there is no definition of Xo'yot in Book vii., and it could equally have been dispensed with here. Similarly De Morgan observes that Euclid never attempts this vague sort of definition except when, dealing with a well-known term of common life, he wishes to bring it into geometry with something like an expressed meaning which may aid the conception of the thing, though it does not furnish a perfect criterion. Thus we may compare the definition with that of a straight line, where Euclid merely calls the reader's attention to the well-known term tiStia ypa^/iij, tries how far he can present the conception which accompanies it in other words, and trusts for the correct use of the term to the axioms (or postulates) which the universal conception of a straight line makes self-evident. We have now to trace as clearly as passible the development of the conception of Xcfyoi, ratio, or relative magnitude. In its primitive sense Xff/os was only used of a ratio between com mensu rabies, i.e. a ratio which could be expressed, and the manner of expressing it is indicated in the proposition, Eucl. x. 5, which proves that commensurate magnitudes have to one another the ratio whieh a numl>er has to a number. That this was the primitive meaning of Aoyoi is proved by the use of the term uAoyoi for the mcom mensurable, which means irrational in the sense of not having a ratio to something taken as rational (^TTot). ^, , , . \t)m I ^^^ BOOK V •!''•' [v. DEF. 3 - 1 Euclid himself shows us how we are to set about finding the ratio, or relative magiiitude, of two commensurable magnitudes. He gives, in x. 3, practically our ordinary method of finding the greatest common measure. If ^, ^ be two magnitudes of which B h the less, we cut off from A a part equal to B, from the remainder a part equal to B, and so on, until we leave a remainder less than B, say Ji,. We measure off ^, frcn S in the same way until a remainder X., is left which is less than fii- We repeat the process with ^1, B,, and so on, until we find a remainder which is contained in the preceding remainder a certain number of times exactly. If account is taken of the number of times each magnitude is contained (with something over, except at the last) in that upon which it is measured, we can calculate how many times the last remainder is contained in A and how many times the last remainder is contained in B ; and we can thus express the ratio of A to B as the ratio of one number to another. But it may happen that the two m^^ttudes have no common measure, i.e. are incommensurable, in which case the process described would never come to an end and the means of expression would fail ; the magnitudes would then Aave na ratio in the primitive sense. But the word Aoyos, ratio, acquires in Euclid, Book v., a wider sense covering the relative magnitude of incommensurabies as well as commensurables ; as stated in Euclid's 4th definition, "magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another," and finite incommensurabies have this property as much as commetisurables. De Morgan explains the manner of transition from the narrower to the wider signification of ratio as follows, "Since the relative magnitude of two quantities is always shown by the quantuplicitative mode of expression, when that is possible, and since pro- portional quantities (pairs which have the same relative magnitude) are pairs which have the same mode {if possible) of expression by means of each other ; in all such cases sameness of relative magnitude leads to sameness of mode of expression ; or proportion is sameness of ratios (in the primitive sense). But sameness of relative magnitude may exist where quantuplicitative expression is impossible ; thus the diagonal of a larger square is the same compared with its side as the diagonal of a smaller square compared with its side. It is an easy transition to speak of sameness of ratio even in this case ; that is, to use the term ratio in the sense of relative magnitude, that word having originally only a reference to the mode of expressing relative magnitude, in cases which allow of a particular mode of expression. The word irraiional (SXsrpsi) does not make any corresponding change but continues to have its primitive meaning, namely, incapable of quantuplicitative expression." It remains to consider how we are to describe the relative magnitude of two incommensurabies of the same kind. That they have a definite relation is certain. Suppose, for precision, that S is the side of a square, D its diagonal ; then, if .S is given, any alteration in D or any error in D would make the figure cease to be a square. At the same time, a person altogether ignorant of the relative magnitude of D and 5 might say that drawing two straight lines of length .S so as to form a right angle and joining the ends by a straight line, the length of which would accordingly be D, does not help him to realise the relative magnitude, but that he would like to know how many diagonals make an exact number of sides. We should have to reply that no number of diagonals whatever makes an exact number of sides ; but that he may mtaition any fraction of the side, a hundredth, a thousandth or a millionth, and that we will then express the diagonal with an error not so great as that fraction. We then teU him that 1,000,000 diagonals exceed T. DW. j] NOTE ON DEFINITION 3 119 1,414,113 sides but fall short of 1,414,214 sides; consequently the diagonal lies between t '41 41 13 and i '4 142 14 times the side, and these differ only by one-millionth of the side, so that the error in the diagonal is less still. To enable him to continue the firocess further, we show him how to perform the arithmetical operation of approximating to the value of J 2. This gives the means of carrying the approximation to any degree of accuracy that may be desired. In the power, then, of carrying approximations of this kind as far as we please lies that of expressing the ratio, so far as expression is possible, and of comparing the ratio with others as accurately as if expression had been possible^ Euclid was of course aware of this, as were probably others before him ; though the actual approximations to the values of ratios of incommensurabies of which we find record in the works of the great Greek geometers are very few. The history of such approximations up to Archimedes is, so far as material was available, sketched in 7%e Works of Archimides (pp. Ixxvti and following); and it is sufficient here to note the facts (i) that Plato, and,e*'en the PythagoreaiK, were familiar with J as an approximfvtion to .j^, {2) that the method of finding any number of successive approximations by the system of side- and iftajfo/taZ-numbere described by Theon of Smyrna was also Pythagorean (cf. the note above on Euclid, n. 9, 10), (3} that Archimedes, without a word of preliminary «tplanation, gives out that gives approximate values for the square roots of several large numbers, and proves that the ratio of the circumference of a circle to its diameter is less than 3t but greater than j-rii (4) '^^t the first approach to the rapidity with which the decimal system enables us to approximate to the value of surds was furnished by the method of sexagesimal fractions, which was almost as convenient to work with as the methoid of decimals, and which appears fully developed in Ptolemy's avyra^vi. A number consisting of a whole number and any fraction was under this system represented as so many units, so many of the fractions which we should denote by ^^, so many of those which we should write (jj)', (A)'> *"<1 ^° ""■ Theon of Alexandria shows us how to extract the square root of 4500 in this sexagesimal system, and, to show how effective it was, it is only necessary to mention that Ptolemy gives -—5 + ^j + ~ as an approximation to ^3, which approximation is equi«ilent to 17320509 in the ordinary decimal notation and is therefore correct to 6 places. Between Def. 3 and Def, 4 two manuscripts and Campanus insert " Pro- portion is the sameness of ratios" (avoAoyta St jJ ruf AoyuiK rnvTonft), and even the best ms. has it in the margin. It would be altogether out of place, since it is not till Def, 5 that it is explained what sameness of ratios is. The words are an interpolation later than Theon (Heiberg, Vol. v, pp. xxxv, Ixxxix), and are no doubt taken from arithmetical works {cf Nicomachus and Theon of Smyrna). It is true that Aristotle says similarly, " Proportion is equality of ratios" (Eth. Nic. v. 6, 1131 a 31), and he appear to be quoting from the Pythagoreans ; but the reference is to numbers. Similarly two mss. (inferiorX insert after Def 7 "Proportion is the similarity {^^juartp) of ratios." Here too we have a mere interpolation. 130 • BOOK V ' • [v. DEFF. 4,5 Definition 4. \iymr ^f(v irpot SXXTjXa. furftdTj Kiytna, A St/rarai inAkiarXiuria^iiiMXi This definition supplements the last one. De Morgan says that it amounts to saying that the magnitudes are of the same species. But this can hardly be aU ; the definition seems rather to be meant, on the one hand, to exclude the relation of a finite magnitude to a magnitude of the same kind which is either infinitely great or infinitely small, and, even more, to emphasise the fact that the term ra/t'e, as defined in the preceding definition, and about to be used throughout the book, includes the relation between any two t'ncom- nunsurable as well as between any two commensurable finite m^nitudes of the same kind. Hence, while De Morgan seems to regard the extension of the meaning of ratio to include the relative magnitude of incommensurables as;, so to speak, taking place between Def. 3 and I>ef. Si the 4th definition appears to show that it is ratio in its extended sense that is being defined in Def. 3- . Definition S' TrfapTOf, otar to, toC TpajTOU KoX TptVov laaKK iroAAairAotrio tw toB Sniripoo xot rtrdpfTov utokk TroKXwrXairitiiv Kau ottoiovovv TroXXairAao'UKr/io^ iKartpcv In my translation of this definition I have compromised between an attempted literal translation and the more expanded version of Simson. 71ie difiiculty in the way of an exactly literal translation is due to the fact that the words (KaS* ijToiovoIi' jroXAaTrXao-iarr/iov) signifying that the equimultiples in eiuh ease are any equimultiples wAa/evcr occur only once in the Greek, though they apply ieiA to Ta....'uTdKK wokkan-kajria in the nominative and Tuv...Uri*is u-oXAmrAno-i'ui' in the genitive. I have preferred "alike " to " simultaneously" as a translation of a/ia because " simultaneously " might suggest that time was of the essence of the matter, whereas what is meant is that any particular comparison made between the equimultiples must be made between (At same equimultiples of the two pairs respectively, not that they need to be compared at the same time, Aristotle has an allusion to a definition of " the same ratio " in Tcfiia VIII. 3, 158 b 29 ; " In mathematics too some things appear to be not easy to prove {ytid^fa6ai) for want of a definition, e.g. that the parallel to the side which cuts a plane [a parallelogram] divides the straight hne [the other side] and the area similarly. But, when the definition is expressed, the said property is immediately manifest ; for the areas and the straight lines Aave the same di'Tai'tn'p«ri9, and this is the definition of 'the same ratio.'" Upon this passage Alexander says similarly, " This is the definition of proportionals which the ancients used : magnitudes are proportional to one another which have (or show) ihe same mSv^tupttrvst and Aristotle has called the latter ivravalpKrit." Heiberg (Mathematisehes zu Aristofeles, p. 2 a) thinks that Aristotle is alluding to the fact that the proposition referred to could not be rigorously proved so long as the Pythagorean definition applicable to com- mensurable magnitudes only was adhered to, and is (quoting the definition belonging to the complete theory of Eudoxus ; whence, m view of the positive statement of Aristotle that the definition quoted is the definition of "the same ratio," it would appear that the Euclidean definition (which Heiberg describes as a careful and exact paraphrase of d-vTovalpftrn) is Euclid's own. I do not V. DEF. s] NOTES ON DEFINITIONS 4, 5 t»l feel able to subscribe to this view, which seems to me to involve very grave difiScuUies. The Euclidean definition is regularly appealed to in Book v. as the criterion of magnitudes being in proportion, and the use of it would appear to constitute the whole essence of the new general theory of proportion; if then this theory is due to Eudoxus, it seems impossible to believe that the definition was not also due to him. Certainly the definition given by Aristotle would be no substitute for it; dvfiv4>aipft7K and dnavoiptaK are words almost a:! vague and " metaphysical " (as Barrow would say) as the words used to define raifa, and it is difficult to see how any mathematical facts could be deduced from such a definition. Consider for a moment the etymology of the words. w^oifKo-ts or dva^ttFii means " removal," " taking away "or " destruction " of a thing; and the prefix om indicates that the "taking away" from one magnitude answers to, corresponds with, alternates with, the " taking away " from the other. So fai» therefore as the etymology goes, the word seems rather to suggest the " taking away " of corresponding fractions, and therefore to suit the old imperfect theory of proportion rather than the new one. Thus Waitz {ad lac.) paraphrases the definition as meaning that " as many parts as are taken from one magnitude, so many are at the same time taken from the other as well," A possible explanation would seem to be that, though Eudoxus had formulated the new definition, the old one was still current in the text-books of Aristotle's time, and was taken by him as being a good enough illustration of what he wished to bring out in the pas.sage of the Ibpia referred to. From the revival of learning in Europe onwards the Euclidean definition of proportion was the subject of much criticism. Campanus had failed to understand it, had in fact misinterpreted it altogether, and he may have misled others such as Ramus {1515 — 72), always a violently hostile critic of Euclid. Among the objectors to it was no less a person than Galileo. For particulars of the controversies on the subject down to Thomas Simpson \Elem. of Geometry, Lond. i8oo) the reader is referred to the Excursus at the end of the second volume of Camerer's Euclid (1825). For us it is interesting to note that the unsoundness of the usual criticisms of the definition was never better exposed than by Barrow. Some of the objections, he pointed out {tttt. Cantabr. vn.ofi665),areduetom isconception onthepartoftheir authors as to the nature of a definition. Thus Euclid is required by these objectors (e.g. Tacquet) to do the impossible and to show that what is predicated in the definition is true of the thing defined, as if any one should be required to show that the name "circle" was applicable to those figures alone which have their radii all equal ! As we are entitled to assign to such figures and such figures only the name of "circle," so Euclid is entitled {"quamvis non temere nee imprudenter at certii de causis iustis illis et idoneis") to describe a certain property which four magnitudes may have, and to call magnitudes possessing that property magnitudes "in the same ratio." Others had argued from the occurrence of the other definition of proportion in vii. Def. so that Euclid was dissatisfied with the present one ; Barrow pointed out that, on the contrary, it was the fact that vu. Def. 3o was not adequate to cover the case of incommensurables which made Euclid adopt the present definition here. Lastly, he maintains, gainst those who descant on the "obscurity" of v. Def. 5, that the supposed obscurity is due, partly no doubt to the inherent difficulty of the subject of incommensurables, but also to faulty translators, and most of all to lack of effort in the learner to grasp thoroughly the meaning of words which, in themselves, are as clearly expressed as they could be. To come now to the merits of the case, the best defence and explanation tss .1' BOOK V tA\-tV [v. DEF. S of the definition that I have seen is that given by De Morgan, He first translates it, observes that it applies equally to commensurable or incom- mensurable quantities because no attempt is made to measure one by an aliquot part of another, and then proceeds thus, "The two questions which must be asked, and satisfactorily answered, previously to its [the definition's] reception, are as follows : 1. What right had Euclid, or any one else, to expect that the preceding most prolix and unwieldy statement should be received by the beginner as the definition of a relation the perception of which is one of the most common acts of his mind, since it is performed on every occasion where similarity or dissimilarity of figure is looked for or presents itself F 2. If the preceding question should be clearly answered, how can the definition of proportion ever be used ; or how is it possible to compare every one of the infinite number of multiples of ji with every one of the multiples of^? To the first question we reply that not only is the test proposed by Euclid tolerably simple, when more closely examined, but that it is, or might be made to appear, an easy and natural consequence of those (iandamental perceptions with which it may at first seem difficult to compare it." To elucidate this De Morgan gives the following illustration. Suppose there is a straight colonnade composed of equidistant columns {which we will understand to mean the vertical lines forming the axes of the columns respectively), the first of which is at a distance from a bounding wall equal to the distance between consecutive columns. In front of the colonnade let there be a straight row of equidistant railings (regarded as meaning their axes), the first being at a distance from the bounding wall equal to the distance between consecutive railings. Let the columns be numbered from the wall, and also the railings. We suppose of course that the column distance (say, C) and the railing distance (say, Ji) are different and that they may bear to each other any ratio, commensurable or incommensurable ; i.e, that there need not go any exact number of railings to any exact number of columns. I \t a 4 B 6 T 8 fl 10 tl 12 la 14 tfl la 17 la If the construction be supposed carried on to any extent, a spectator can, by mere inspection, and without measurement, compare C with Ji to any degree of accuracy. For example, since the loth railing falls between the 4th and 5th columns, 10^ is greater than 4C and less than $C, and therefore Jl lies between -yVhs of C and yjjths of C. To get a more accurate notion, the ten-thousandth railing may be talcen ; suppose it falls between the 4674th and 4675th columns. Therefore io,ooo.ff lies between 46 74 C and 4675 C, or ^ hes between tVuVTy ^'^^ rVtyVs ^^ ^' There is no limit to the degree of accuracy thus obtainable ; and the ratio of ^ to C is determined when the order of distribution of the railings among the columns is assigned arf infinitum ; or, in other words, when the position of any giver railing can be found, as to the numbers of the columns between which it lies. Any alteration, however small, in the place of the first railing must at last affect the order of distribution. Suppose e.g. that the first railing is moved from the wall by one part in a thousand of the distance between the columns ; then the second railing is pushed forward by x7n(irC, the third by nnnr^i and so on, so that V. DEF, 5] NOTE ON DEFINITION 5 i«3 the railings after the thousandth are pushed forward by more than C; i.e. the order with respect to the columns is disarranged. Now let it be proposed to make a model of the preceding construction in which c shall be the column distance and r the railing distance. It needs no definition of proportion, nor anything more than the conception which we have of that term prior to definition {and with which we must show the agree- ment of any definition that we may adopt), to assure us that C must be to J in the same prof>ortion as ^ to r if the model be truly formed. Nor is it drawing too largely on that conception of proportion to assert that the distribution of the railings among the columns in the model must be every- where the same as in the original ; for example, that the model would be out ef proportion if its 37th railing fell between the i8th and 19th columns, while the 37th rathng of the original fell between the 17th and iSth columns. Thus the dependence of EucHd's definition upon common notions is settled; for the obvious relation between the construction and its model which has just been described contains the collection of conditions, the fulfilment of which, according to Euclid, constitutes proportion. According to Euclid, whenever mC exceeds, equals, or falls short of nR, then tttc must exceed, equal, or fall short of nr; and, by the most obvious property of the constructions, according as the wth column comes after, opposite to, or before the nth railing in the original, the ffith column must come after, opposite to, or before the ffth railing in the correct model. Thus the test proposed by Euclid is necessary. It is also sufficient. For admitting that, to a given original with a given column-distance in the model, there is one correct model railing distance (which must therefore be that which distributes the railings among the columns as in the original), we have seen that any other railing distance, however slightly different, would at last give a difTerent distribution ; that is, the correct distance, and the correct distance only, satisfies all the conditions required by Euclid's definition. The use of the word diitribtition having been well learnt, says De Morgan, the following way of stating the definition will be found easier than that of Euclid. " Four magnitudes, A and B of one kind, and C and D of the same or another kind, are proportional when all the multiples of A can be distributed among the multiples of Bm the same intervals as the correspond- ing multiples of C among those of D." Or, whatever numbers m, n may be, if mA lies between ttB and {n + i)B, mC lies between nD and (« + i)£>. It is important to note that, if the test be always satisfied from and after any given multiples of A and C, it must be satisfied before those multiples. For instance, let the test be always satisfied from and after \oaA and looC; and let f)A and 5C be instances for examination. Take any multiple of 5 which will exceed 100, say 50 times five ; and let it be found on examination that 250^4 lies between 678^ and 67g.fi ; then 150 c lies between 678ZJ and f>1^D. Divide by 50, and it follows that ^A lies between \%\%B and tzWB, and ajortiori between i^B and \^B. Similarly, 5 dies between isl^^Z? and 13JJ/), and therefore between \%D and 14Z?. Or ^A lies in the same interval among the multiples of B in which 5 C lies among the multiples of D, And so for any multiple of A, C less than 100^, looC. There remains the second question relating to the infinite character of the definition ; four magnitudes A, B, C, D are not to be called proportional until it \b shown that every multiple of A falls in the same intervals among the multiples of 5 in which the same multiple of C is found among the multiples of D. Suppose that the distribution of the raihngs among the '^''' BOOK V [v. DEF, S columns should be found to agree in the model and the original as far as the millionth railing. This proves only that the railing distance of the model does not err by the millionth part of the corresponding column distance. We can thus fix limits to the disproportion, if any, and we may make those limits as small as we please, by carrying on the method of observation; but we cannot obscrue an infinite number of cases and so enable ourselves to affirm proportion absolutely. Mathematical methods however enable us to avoid the difficulty. We can take any multipks whatever and work with them as if they were particular multipJes. De Morgan gives, as an instance to show that the definition of proportion can in practice be used, notwithstanding its infinite character, tiie following proof of a proposition to the same effect as EucL VI. 3. o. A, oj At "Let OAB be a triangle to one side AB of which ab is drawn parallel, and on OA produced set off A At, A^Af etc. equal to OA, and aa^ a,£i, etc. equal to Oii. Through every one of the points so obtained draw parallels to AB, meeting OB produced in b^, B, etc. Then it is easily proved that W„ bj^, etc. are severally equal to Ob, and BB^, B^Bi etc. to OB. Consequently a distribution of the multiples of OA among the multiples of Oa is made on one line, and of OB among those of Ob on the other. The examination of this distribution in all its extent (which is impossible, and hence the apparent difficulty of using the definition) is rendered unnecessary by the known property of parallel lines. For, since At lies between a, and a„ B^ must he between b^ and ^,j for, if not, the line A^B^ would cut either a^, or a^^. Hence, without inquiring where A,^ doei fall, we know that, if it fall between a, and a,^„ B,^ must fall between b„ and ^,+1 ; or, if m . OA fall in magnitude between n.Oa and (n + i)(?a, then m.OB must fall between n.Ob and («+i)Oi." Max Simon remarks {Euclid und die seeks planimeirischen Buchtr, p. no), after Zeuthen, that Euclid's definition of equal ratios is word for word the same as Weierstrass' definition of equal numbers. So far from agreeing in the usual view that the Greeks saw in the irrational no number, Simon thinks it is clear from Eucl. v. that they possessed a notion of number in all its generality as clearly defined as, nay almost identical with, Weierstrass' con- ception of it Certain it is that there is an exact correspondence, almost coincidence, between Euclid's definition of equal ratios and the modern theory of irrationals due to Dedekind. Premising the ordinal arrangement of natural numbers in ascending order, then enlarging the sphere of numbers by including (i) negative numbers as well as positive, (2) fractions, as ajb, where a, b may V. DEF. i] NOTE ON DEFINITION s itj be any natural numbers, provided that i is not zero, and arranging the fractions ordinally among the other numbers according to the definition : let 1 be < = > J according as a;/ is < = > ii:, e a Dedekind arrives at the following definition of an irrational number. An inatianal number a is defined whenever a law is stated which will assign every given rational number to one and only one of two classes A and B such that (i) every number in A precedes every number in j5, and (2) there is no last number in A and no first number in B ; the definition of a being that it is the one number which lies between all numbers in A and all numbers in B. Now let xly and ar'/y be equal ratios in Euclid's sense. Then ~ will divide all rational numbers into two groups A and B ; —, „ „ „ A' and B". Let -; be any rational number in A, so that tax ,. This means that ay <bx. But Euclid's definition asserts that in that case af-cbs! also. Hence also 7 < -1 ; b y therefore every member of group A is also a member of group A'. Similarly every member of group ^ is a member of group B". For, if T belong to £, ax which means that ay > bx. But in that case, by Euclid's definition, «y > bx' ; therefore also i> -j- y Thus, in other words, A and B are coextensive with A' and S respectively ; therefore - = — , according to Dedekind, as well as according to Euclid. If x(y, :^iy happen to be rational, then one of the groups, say A, includes xjy, and one of the groups, say A', includes x'jy'. ' • d . X •■!.,-"' In this case r might mncide with - ; ,..-., X that is r = - 1 b y' which means that ay^bx. i«6 ' '^^ BOOK V •' ' [v.DW. S Therefore, by Euclid's definition, ay = &e' ; -i-'wi-'* timu .,i.. •-■ SO that T^-5. y Thus the groups are again coextensive. In a woid, Euclid's definition divides all rational numbers into two coextensive classes, and therefore defines equal ratios in a manner exactly corresponding to Dedekind's theory. Alternativea for Eucl. V. Dcf. 5. Saccheri records in his Evclides ob omni noivo vindicatus that a distinguished geometer of his acquaintance proposed to substitute for Euclid's the following definition : "A first magnitude has to a second the same ratio that a third has to a fourth when the first contains the aliquot parts of the second, auording to any number [i.e. with any denominator] whatever, the same number of times as the number of times the third contains the same aliquot parts of the fourth " ; on which Saccheri remarks that he sees no advantage in this definition, which presupposes the notion of division, over that of Euclid which uses multiplication and the notions o{ greater, equal, and less. This definition was, however, practically adopted by Faifofer [Elementi it geometria, 3 ed., iSSsi) in the following form 1 " Four infinitudes taken in a certain order form a proportion when, by measuring the first and the third respectively by any equi-submultiples whatever of the second and of the fourth, equal quotients are obtained," Ingrami {Elementi di geometria, 1904) takes multiples of the first and third instead of submultiples of the second and fourth : " Given four magnitudes in predetermined order, the first two homogeneous with one another, and likewise also the last two, the magnitudes are said to form a proportion (or to be in proportion) when any multiple of the first contains the second the same number of times that the equimultiple of the third contains the fourth." Veronese's definition {Elementi di geometria, PL 11., 1905) is like that of Faifofer; Enriques and Amaldi {Elemtnti di geom^ria, 1905) adhere to Euclid's: Proportionals of VII. Def. ao a particular case. It has already been observed that Euclid has nowhere proved (though the fact cannot have escaped him) that the proportion of numbers is included in the proportion of magnitudes as a special case. This is proved by Sim son as being necessary to the 5th and 6th propositions of Book x. Simson's proof is contained in his propositions C and D inserted in the text of Book v, and in the notes thereon. Proposition C and the note on it prove that, if four magnitudes are proportionals according to vii. Def. 20, they are also proportionals according to v. Def. 5. Prop, D and the note prove the partial converse, namely that, if four magnitudes are proportionals according to the Sth definition of Book v., and if the first be any multiple, or any part, or parts, of the second, the third is the same multiple, part, or parts, of the fourth. The proofs use certain results obtained in Book V. Prop. C is as follows ; If the first be the same multiple of the second, or the same part of it, that tht third is qfthe fourth, the first is to the second as the third to the fourth. V. DEF. s] NOTE ON DEFINITION s nf Let the (list A be the same multiple of B the second that C the third is of the fourth D \ ^ is to ^ as C is to 27. A e B O C F D H ■ ' Take of A^ C any equimultiples whatever E, F\ and of B, D any equimultiple whatever G, H. Then, because -4 is the same multiple of B that C is of 27, 'i- " ; '" and E is the same multiple o{ A that F\% of C, E is the same multiple of B that F\% of D. [v, 3] Therefore E, E&k the same multiples of B, D. But G, H d,K equimultiples of ^, D; therefore, if £ be a greater multiple of B than G is, F'\% a greater multiple of JD than ^ is of i? ; that is, if £ be greater than G, Fis greater than If. In like manner, if E be equal to G, or less, ^is equal to J/, or less than it. But E, Fate equimultiples, any whatever, of ^, C; and G, H any equimultiples whatever of B, D. Therefore ^ is to ^ as C is to Z>. [v, Def. 5] Next, let the first A be the same pari of the second B that the third C is of the fourth D ; j4 is to i? as C is to ZJ. A For B is the same multiple of A that 2> is of C; B wherefore, by the preceding case, q ^ is to j4 as /J is to C; O and, imxrsefy, A is to B as C is to D. [For this last inference Sirason refers to his Proposition B. That proposition is very simply proved by taking any equimultiples E, F of B, Z) and any equimultiples G, Hoi A, C and then arguing as follows : Since A\%to B && C is to Z>, G, If are simuUamously greater than, equal to, or less than E, F respectively ; so that E, F are HmultantauUy less than, equal to, or greater than G, H respectively, and therefore [Def. 5] ^ is to ^4 as i? is to C] We have now only to add to Prop. C the case where AB contains the same parts of CD that EFAaes of GH: in this case likewise AB is to CD as EFto GIf. Let CJC be a part of CD, and GL the same part of GIf; let AB be the same multiple of CAT that EFis of GL. iaS ' ^'- BOOK V [v. T>KF. 5 Therefore, by Prop. C, '■"■ ■ '"• — - ■ A£ is to C^as £Fu> GL, B E- G- c— R ^ And CD, GH ire equimultiples of CK, GL, the second and fourth. Therefore AB is to CD as EF to G^ [Simson's Cor, to v. 4, which however is the particular case of V. 4 in which the " equimultiples " of one pair are the pair itself, i.e. the pair multiplied by unity]. To prove the partial converse we begin with Prop. D. If the first be to the second as the third to the fourth, and if the first be a multiple or part of the second, the third is the same mulliple or the same part of the fourth. Let v4 be to ^ as C is to i> ; and, first, let /i be a multiple of B ; C is the same multiple of S. Take E equal to A, and whatever multiple A 01 E \s of B, malce F the same multiple of Z>. Then, because A is Xo B a& C\s 10 D, <■' and of B the second and D the fourth equimultiples have been taken E and F, /i is to £ as C is to ^ [v. 4, Cor.] But A is equal to E ; therefore C is equd to F. [In support of this inference Simson cites his Prop. A, which however we can directly deduce from v. Def. 5 by taking any, but the same, equimultiples of all four magnitudes.] A C- B D- e— F- Now ,^is the same multiple of Z> that Aisol B; therefore C is the same multiple of D that A is of B. Next, let the first /i be a part of the second B ; C the third is the same part of the fourth D, Because ^i is to ,5 as C is to D, inversely, J is to -4 as Z) is to C. [Prop. B] But A'y&a. part of J; therefore ,5 is a multiple of >€; and, by the preceding case, D is the same multiple of C, that is, C is the same part of D that A is of B. We have, again, only to add to Prop D the case where AB contains any parts of CD, and AB is to CD as EFio GH; then shall EF zontain the same parts of GB that AB does of CD. ?■ DWr. 5—7] NOTES ON DEFINITIONS g— 7 199 For let CIC be a part of CJ), and G'Z the same part of Gff; and let j4£ be a multiple of CX. jff^ shall be the same multiple of GL. Take M the same multiple of GL that AB h of CA"; therefore ji£ is to CAT as ^ is to GL. [P"*p. C] A B E — F C tr- G- L H M And CI>, G/f&te equimultiples of CA", GL; therefore /4B is to CD as j}/ is to GIL But, by hypothesis, A£ is to CZJ as £^is to GB"; therefore M is equal to £J\ [v. 9] and consequently £^i3 the same multiple of GL that A£ is of Cff. Definition 6. ' t ■ Tb Si TOV airoy Ij^dktu \6yov /itye&Tj avaXjoyov KoXturBa, x > 'A,vdXoryoy, though usually written in one word, is equivalent to aVi Xoyot', /» proportion. It comes however in Greek mathematics to be used practically as an indeclinable adjective, as here ; cf. oi Tto-o-opt? euflttdi avaktyfov laovtiu., " the four straight lines will be proportional," tflyw/a. rat irXm/Mi aVoAoyov Ix"**^ "triangles having their sides proportional." Sometimes it is used adverbially : akoXoyov o^ k^iv ci>c 17 BA irpo? r^v AF, outq^^ ?^ HA irpo^ T^f AZ, "proportionally therefore, as BA is to AC, so is CZJ to £>F"\ so too, ap- parently, in the expression ij /lAnj aniXoyoi/ (*ufl<ut), " the mean proportional." I do not follow the objection of Max Simon (Euclid, p. no) to "proportional" as a translation of oniAoyoi'. "We ask," he says, "in vain, what is proportional to what? We say e.g. that weight is proportional to price because double, treble etc. weight corresponds to double, treble etc. price. But here the meaning must be 'standing in a relation of proportion.'" Yet he admits that the Latin word preportionalis is an adequate expression. He transl.ttes by "in proportion" in the text of this definition. But I do not see that "in proportion " is better than "proportional." The fact is that both expressions are elliptical when used of four magnitudes " in proportion " ; but there is surely no harm in using either when the meaning is so well understood. The use of the word naXtiV^, " let magnitudes having the same ratio be called proportional," seems to indicate that this definition is Euclid's own. Definition 7. TOv Tov Btvripov iroAXa^rXaatou, to Sc to€ rpirov TroWaTrKa^ior ^ij {ijrtpi)fff rov ro3 rtropTou troXXavXairltni, rott to rpajrov irpoi to Stvrtpor fWifuwi Xoyoi" •X'"' Xfycrat, ^tp ra rpiVcn^ vpo^ to TtToprov. As De Moi^an observes, the practical test of disproportion is simpler than that of proportion. For, whereas no examination of individual cases, however 130 ■ BOOK V ' [v. DBF. 7 extensive, will enable an observer of the construction and its model (the illustration by means of columns and railings described above) to affinn proportion or deny disproportion, and all it enables us to do is to fix limits (as small as we please) to the disproportion (if any), a single instance may enable us to deny proportion or affirm disproportion, and also to slate which way the disproportion lies. Let the igth railing in the original fall beyond the nth column, while the 15th railing of the (so-called) model does not come up to the nth column. It follows from this one instance that the railing distance of the model is too small relatively to the column distance, or that the column distance is too great relatively to the railing distance. That is, the ratio of /• to r is less than that of /? to C, or the ratio of f to r is greater than that of C to jR. Saccheri (<7/. at,) remarks (as Commandinus had done) that the ratio of the first magnitude to the second will also be greater than that of the third to the fourth if, while the multiple of the first is efuai to the multiple of the second, the multiple of the third is Uss than that of the fourth : a case not mentioned in Euclid's definition. Saccheri speaks of this case being included in Clavius' interpretation of the definition. 1 have, however, failed to find a reference to the case in Clavius, though he adds, as a sort of corollary, in his note on the definition, that if, on the other hand, the multiple of the first is iess than the multiple of the second, while the multiple of the third is nt>i las than that of the fourth, the ratio of the first to the second is kss than that of the third to the fourth. Euclid presumably left out the second possible criterion for a greater ratio, and the definition of a less ratio, because he was anxious to reduce the definitions to the minimum necessary for his purpose, and to leave the rest to be inferred as soon as the development of the propositions of Book v. enabled this to be done without difficulty. Saccheri tried to reduce the second possible criterion for a greater ratio to that ^ven by Euclid in his definition without recourse to anything coming later in the Book, but, in order to do this, he has to use "multiples" produced by multipliers which are not integral numbers, but integral numbers //uJ proper fractions, so that Euclid's Def. 7 becomes inapplicable. De Morgan notes that " proof should be given that the same pair of magnitudes can never offer both tests [i.e. the test in the definition for a greater ratio and the corresponding test for a less ratio, with "less" substituted for "greater" in the definition] to another pair; that is, the test of greater ratio from one set of multiples, and that of less ratio from another." In other words, if m, n, p, q are integers and A, B, C, D four magnitudes, none of the pairs of equations (i) mA->HB, mC=ai <nD, (a) mA = nB, mC < nD can be satisfied simultaneously with any one of the pairs of equations (3) pA^qB, pC>qD, (4) pA < qB, y*C > or = qD. There is no difficulty in proving this with the help of two simple assumptions which are indeed obvious. We need only take in illustration one of the numerous cases. Suppose, if possible, that the following pairs of equations are simultaneously true : (l) viA>nB, mC<nD and (2) pA <qB, pC>qD. V. DEFF. r, 8] NOTES ON DEFINITIONS ;, 8 iji Multiply (i) by q and (z) by n. (We need here to assume that, whet« rX, rK are any equimultiples of any magnitudes X^ ¥, according as X>- = < Y, rX> = <rY. This is contained in Simson's Axioms i and 3.) We have then the pairs of equations my A > ngB, mqC < n^D, npA<nqB, npC>nqD, From the second equations in each pair it follows that mqC < npC. (We now need to assume that, if rX, sX are any multiples of X, and rY, sY the same multiples of Y, then, according as rX >-< sX, rY-> = < sY. Simson uses this same assumption in hb proof of v. i3.) Therefore mqA <npA, ■ , But it follows from the first equations in each pair that mqA > npA : which is impossible. Nor can Euclid's criterion for a greater ratio coexist with that for equal ratios. Definition 8. ^ kvoXvfia, Si iv fptaXv opoit ISaxurryi hrriv. This is the reading of Heiberg and Camerer (who follow Peyrard's Ms,) and is that translated above. The other reading has Aaxiorcni, which can only be translated "consists in three terms a/ hast." Hankel regards the defi- nition as a later interpolation, because it is superfluous, and because the word ojxK for a term in a proportion is nowhere else used by Euclid, though it is common in later writers such as Nicomachus and Theon of Smyrna. The genuineness of the definition is however supported by the fact that Aristotle not only uses Spot in this sense {Eth. Nic. v. 6, 7, 1131 b 5, 9), but has a similar remark {ibid. 1131 a 31) that a "proportion is in fQiir terms at least" The difference from Euclid is only formal ; for Aristotle proceeds : " The diicrett (Si^p^fiini) (proportion) is clearly in four (terms), but so also is the continuous (awtxv'). For it uses one as two and mentions it twice, e.g. (in stating) that, as a is to j3, so also is j3 to y ; thus j3 is mentioned twice, so that, if /9 be twice put down, the proportionals are four." The disrinction between discrete and C07ttinuous seems to have been Pythagorean (cf. Nicomachus, 11. 11, 5; 23, a, 3; where however o-un/^fioTj is used instead of Tvvtxq^); Euclid does not use the words Stjpit|/t(n; and awtx^i in this connexion. So far as they go, the first words of the next definition (g), "When three magnitudes are proportionals," which seemingly refer to Def. 8, also support the view that the latter is, at least in substance, genuine. ij» *' BOOK V -'•:■' [v. DKFF. 9,10 <■' •••••- Definitions 9, la t 9. 'Oral' Si Tpi'a fityi^rj ivaXayov % to irpwroi' jrpos ri TpiTOf fcirXnirtova Xoyov <;(<£i^ Xcycrai tttc/i irput to BtvTtpov. 10. 'Orair Sc THTO'a^ML fttyi&Tq aVaXoyoi^ ^^ to vfitarav irpm To rireifrTOV Tpurkatrlova Awyoy fx** '^ty^riM ^irep irpos li MvTtpoy, nai dtt if^t opHuii, wt Here, and fn connexion with the definitions of duplicate, triplicate, etc. ratios, would be the place to expecta definition of "compaunii ratio." None such is however forthcoming, and the only "definition" of it that we find is that forming vi. Def. Si which is an interpolation made, perhaps, even before Theon's time. According to the interpolated definition, " A ratio is said to be compounded of ratios when the sizes (jnfAtKo'nTret} of the ratios multiplied together make some (? ratio)." But the multiplication of the li^s (or magnitudes) of two ratios of incommensurable, and even of commensurable, magnitudes is an operation unknown to the classical Greek geometers. Eutocius (Archimedes, ed. Heiberg, iii. p. lao) is driven to explain the definition by making irijAwonj^ mean the number from which the given ratio is called, or, in other words, the number which multiplied into the consequent of the ratio gives the antecedent. But he is only able to work out his idea with reference to ratios between numbers, or between commensurable magnitudes ; and indeed the definition is quite out of place in Euclid's theory of proportion. There is then only one statement in Euclid's text as we have it indicating what is meant by compound ratio ; this is in vi: 23, where he says abruptly "But the ratio of KXa M is compounded of the ratio of JT to Z and that of L to M." Simson accordingly gives a defitiition (A of Book v.) of compound ratio directly suggested by the statement in vi. 23 just quoted. " When there are any number of magnitudes of the same kind, the first is said to have to the last of them the ratio compounded of the ratio which the first has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and so on unto the last magnitude. For example, if A^ B, C, D b& four ma^itudes of the same kind, the first A is said to have to the last D the ratio compounded of the ratio of A to B, and of the ratio of B to C, and of the ratio of C to Z) ; or the ratio o^ A 10 Dm said to be compounded of the ratios of A ta S, B to C, and C to D. And if .(4 has to .ff the same ratio which E has to F; and .ff to C the same ratio that G has to /f ; and C to Z> the same that A" has to L; then, by this definition, A is said to have to D the ratio compounded of ratios which are the same with the ratios of E io F, G to H, and A" to Z : and the same thing is to be understood when it is more briefly expressed, by saying, A has to D the ratio compounded of the ratios of £■ to ^ G to If, and JT to Z, In like manner, the same things being supposed, if M has to N the same ratio which A has to D ; then, for shortness' sake, M is said to have to A^the ratio compounded of the ratios of -£■ to ^ G to H, and A" to L." De Morgan has some admirable remarks on compound ratio, which uot only give a very clear view of what is meant by it but at the same time V. vwr. 9, lo] NOTES ON DEFINITIONS 9, 10 133 supply a. plausible explanation of the origin of the term. "Treat ratio," says De Morgan, ''as an engine of operation. Let that of j4 to .ff surest the power of altering any magnitude in that ratio." (It is true chat it is not yet proved that, B being any magnitude, and /" and Q two magnitudes of the same kind, there does exist a magnitude ji which is to JS in the same ratio OS /* to Q. It is not till vi. 1 3 that this is proved, by construction, in the particular case where the three magnitudes are straight lines. The proof in the Greek text of v, 18 which assumes the truth of the more general proposition is, by reason of that assumption, open to objection ; see the note on that proposition.) Now "every alteration of a magnitude is alteration in some ratio, two or more successive alterations are jointly equivalent to but one, and the ratio of the initial magnitude to the terminal one is as properly said to be the compound ratio of alteration as 13 to be the compound addend in* lieu of 8 and 5, or 28 the compound multiple for 7 and 4. Competition is used here, as elsewhere, for the process of detecting one single alteration which produces the joint effect of two or more. The composition of the ratios of P \a R, R \.o S, 7" to 6^ is performed by assuming A, altering it in the first ratio into B, altering B in the second ratio into C, and C in the third ratio into D. The joint effect turns A into D, and the ratio o( A to D is the compounded ratia" Another word for (ompouncUd ratio is crvn^jK^wot (cruraTmu) which '\% common in Archimedes and later writers. It is clear that diiplicate ratio, triplicate ratio etc. defined in v. Deff. 9 and 10 are merely particular cases of compound ratio, being in fact the ratios compounded of two, three etc tqual ratios. The use which the Greek geometers made of compounded, duplicate, triplicate ratios etc. is well illustrated by the discovery of Hippocrates that the problem of the duplication of the cube (or, more generally, the construction of a cube which shall be to a given cube in any given ratio) reduces to that of finding "two mean proportionals in continued proportion." This amounted to seeing that, if X, y are two mean proportionals in continued proportion between any two lines a, b, in other words, if a is to jc as a; to^, and a: is toj* as jf to #, then a cube with side a is to a cube with side .x as a is to b\ and this is equivalent to saying that a has to b the triplicate ratio of a to .-t*. Euclid is careful to use the forms SorXatri'iuK, tpntXaaiioy, etc. to express what we translate as dupiicait, tripliatte etc. ratios ; the Greek mathematicians, however, commonly used StirXoffios Xoyg^, "double ratio," TpiirAao-ios Xoyw, "triple ratio " etc, in the sense of the ratios of i to i, 3 to i etc. The effort, if such it was, to keep the one form for the one signification and the other for the other was only partially successful, as there are several instances of the contrary use, e,g. in Archimedes, Nicomachus and Pappus. The expression for having the ratio which is " duplicate (triplicate) of that which it has to the second" is curious — S«rAacrio™ (Tfjin-Aao-i'oi'a) \irfov Ixtw ^rcp 7rp« TO ieuTt^r — -^Trtp being used as if SiwXafftova or TpiTrXtwtoi'fl were a sort of comparative, in the same way as it is used after ftfifom or tAno-o-ovo. Another way of expressing the same thing is to say XiJyiK Sn-Xatrtwi' {^fsatXasrlani) TO if, %v lyy.... the ratio "duplicate of that (ratio) which,,." The explanation of both constructions would seem to be that StirXoo-io; or S(7rAacrw»' is, as Hultsch translates it in his edition of Pappus (cf p. 59, 17), duplo maior, where the ablative duplo implies not a difference but a proportion. The four magnitudes in Def, 10 must of course be in continued proportion (Kara to <rv>'(;(n). The Greek text as it stands does not state this. t^ o; . BOOK V "^'-nf [v. DEFF, II— 14 v'..-.;«iai^.r'- m DEFINITION II. "' ''•■" '0/M\oya fuyi^Ti KiyiTot To fihi TJytyv/ittva Toij tf)N)ti/i(ViH9 ra Si jira;ic)« Tott It is difficult to expre^ the meaning of the Greek in as few words. A translation more literal, but conveying less, would be, "Antecedents are called wrrtsponding magnitudes to antecedents, and consequents to consequents." I have preferred to translate i^Xoyo* by " corresponding " rather than by " homologous." I do not agree with Max Simon when he says (Euclid, p. 1 11) that the technical term "homologous" is not the adjective i/uSAnyo!, and does not mean "corresponding," "agreeing," but "like inrespect of the proportion" ("ahnlich in Bezug auf das Verhaltniss"), The definition seems to me to be for the purpose of appropriating to a technical use precisely the ordinary adjective A/ioAoyos, "agreeing" or "corresponding." Atttt(tdtnts, TJyov/JLtva, are literally "leoiiing (terms)," and cotistijuents , iini/itvo, "following (terms)." Definition 12. "EvaAAoi Aifyo* foTt Xiji/fii toB jj-you/atrou irpos To -tfytti^tyvi/ mi tou hm^vow irpoc TO hm^vGV^ We now come to a number of expressions for the transformation of ratios or proportions. The first is ifoAAiif, alUrnaldy, which would be better described with reference to a proportion of four terms than with reference to a ratio. Bui probably Euclid defined all the terms in DeiT. li — 16 with reference to ratios because to define them with reference to proportions would look like assuming what ought to be proved, namely the legitimacy of the various transformations of proportions (cf. v, 16, 7 Por., 18, 17, 19 For,). The word iraXXof is of course a common term which has no exclusive reference to mathematics. But this same use of it with reference to proportions already occurs in Aristotle: Anal. post. t. 5, 74 a 18, i«u *o dvoAcryoi' ort ^™AAi»f, "and that a proportion (is true) alternately, or alttrnando" Used with Aoyos, as here, the adverb ivaXka^ has the sense of an adjective, "alternate"; we have already had it similarly used of " alternate angles " (at JvoAAof yuiruu) in the theory of parallels. Definition 13. AvamAu' Aayot itrrX A^^ii tou iiro/io'ou wc vyov/nrwiu irpo? to ^ytmiitvm/ wc Jro/MVOf. 'AvdiraXiv, " inversely," " the other way about," is also a general term with no exclusive reference to mathematics. For this use of it with reference to proportion cf, Aristotle, De Cado i. 6, i73b32TiJi' ovaAoyiV ^v to jSiipTj Ix", oE xpwoi avatrakai tfoiwiv, " the proportion which the weights have, the times will have invtrsely." As here used with Aoyo?, avdvaXw is, exceptionally, adjectival. Definition 14. "Xwvtfri^ Aoyou i<n\ Xi^^4.f TOtJ ^jyavfmivov fttri row iTrofj^fvov <i>s if os wpo* auTo TO trontfav. The tomfosition of a ratio is to be distinguished from the cetnpounding of ratios and compounded ratio {<rvyttiijifvo% Aoyoi) as explained above in the note ». DKTF. 14— 16] NOTES ON DEFINITIONS ii— 16 133 on Deff. 9, ro. The fact is that oTPrntftj/it and what serves for the passive of it (ffvymtfww) are used for adding as well as compounding in the sense of compounding ratios. In order to distinguish the two senses, I have always used the word wmpomndo where the sense is that of this definition, though this requires a slight departure from the literal rendering of some passages. Thus the enunciation of v. 17 says, literally, "if magnitudes compounded be in proportion they will also be in proportion separated" (Wv crvyitttfMfa /jwytAj ixiktrjov ^, Kol Sto(p«6fl/To omAoyov fo-rai). This practically means that, if j4 + 5 is to .S as C + Z) is to A then v4 is to J as C is to D. I have accordingly translated as follows : " if magnitudes be proportional eompomnde, they will also be proportional separando" (It will be observed that stparattdo, a term explained in the next note, is here used, not relatively to the proportion /(! is to .ff as C is to D, but relatively to the proportion compenmdo, viz. A -^ B \& xo B as, C + Z* is to 2).) The corresponding term for eomponendo in the Greek mathematicians is avvBivn, literally "to one who has compounded," i.e. " if we compound." (For this absolute use of the dative of the participle cf Nicomachus i. 8, 9 otto /ioi'<iSijt...™ia tov SurXaa-ioc Xdyoi' jrpojfiupouvTi ni)(pK Aimpow, hoi ictu &v yiviiivTat, ovrot Travrn &pTWiKK apTioi tiinv. A very good instance from Aristotle is £(A. Ni(. 1. 5, 1097 b la itttt^T^ivovTK yap ^4 roii^ yokcis koX tous diroyovovi koX rw f^ikattf roix ^lXau^ tK arfifiov wfxifunv.) A variation for mivSivri, found in Archimedes is nara <rvi/$nTLv, Perhaps the more exclusive use of the form <rur0ivTi by geometers later than Euclid to denote the composition of a ratio, as compared with Euclid's more general use of mvStaxa and other parts of the verb truirtftjfu or ovymifuu, may point to a desire to get rid of ambiguity of terms and to make the terminology of geometry more exact. Definition 15. Ataip«r(s Xoyov itm. Xifi^i^ r^v inrt^^i, ^ vvt^pi-^L to i^you^cpoi^ tov kico^vvfOy TTpb^ (XvTO rh hrofAivov, As composition of a ratio means the transformation, e.g., of the ratio of A Xa B into the ratio oi A + B to S, so the uparcUion of a ratio indicates the transformation of it into the ratio oi A~ B to B. Thus, as the new antecedent is in one case got by adding the original antecedent to the original consequent, so the antecedent in the other case is obtained by subtracting the original consequent from the original antecedent (it being assumed that the latter is greater than the former). Hence the literal translations of tiaifnirn Xoyou, "division of a ratio," and of htXovti (the corresponding term to <rvr6ivri) as dividtndo, scarcely give a sufficiently obvious explanation of the meaning. Heiberg accordingly translates by "subtractio lationis," which again may be thought to depart too far from the Greek. Perhaps "separation" and separando may serve as a compromise. ■ *.ir%-»*r!\ jnj Definition 16. "AKotTTpo^ Ao'yoii /<rTt AiJ^is roG ijyoufiA'Oii nrpos njc vrtpoxijV, ^ ■Airtpiyn Conversion of a ratio means taking, e.g., instead of the ratio of A to B, the ratio oi A to A —JB {A being again supposed greater than B). As iwwTpo^i7 is used for conversion, so ivaxTtpiiftavri is used for conver tends (corresponding to the terms avii$ivtt, and SwAdvrt). I36 ■■ BOOK V '• ' [v. DEfF. 17, 18 Definition 17. trivia Xafi^avQfiivttfr nal iv T(3 avrcp Aoyitij orav ^ ii^ iy rol^ wp^-rotv fuytOttrt to vpwTOr itpm TO itr}(aToy, outuie ivTott Sivripoit fiTytSiiTno irfKuTOk' *poi to JtrjjaToi' ■ ^ oAXuT' \^>frK T)a¥ axpuiv koB' vw4fyitpt<n.r Tmv jiiam>. Si tcrov, tx atquali, must apparently mean ex atquali dislaniia, at an equal distance or interval, i.e. after an equal number of intervening terms. The wording of the definition suggests that it is rather a proportion ex atquali than a ratio ex aet/uali which is being defined (cf. Def. 12). The meaning is clear enough. If a, 6,c,d...ht one set of magnitudes, and A, B, C, D... another set of magnitudes, such that (I is to * as vi is to B, ^ is to f as .d is to C, and so on, the last proportion being, e.g., /' is to /, as A* is to Z, then the inference ex aeqtmli is that a is to / as /4 is to i. ' The/orf that this is so, or the truth of the inference from the hypothesis, is not proved until v. 22. The definition is therefore merely verbal; it gives a convenient name to a certain inference which is of constant application in mathematics. But ex aequali could not be intelligibly defined except with reference to two sets of ratios respectively equal. , Definition 18. TtTopayjiivTi Si A.vaXoyia ttniv^ orav t^loiv ^VTtffV fMytS^ ircu aXAai»' aurotc r<r«i' TO Trk^ftov yiVip-ai ut /xiv iv to« jfpcoToit fHyiSiirtv^oviityoir irpos Irofitvov, ovT4i>c iy Tots otvTtpot.^ fiiyi&tiTtv ^tivfitvtty wpo^ Iwofi-tvov, mi Si iv rots Tr/xJroi^ lktyiBi<rKV inop^fvoy irpos aXka Tc, ourcii? iy this £<UTCpot; aXXo Tt vphi ^yovfttvov^ Though the words Bi' 'urav, ex atqiitUt, are not in this definition, it gives a description of a case in which the inference ex aeguali is still true, as will be hereafter proved in v. 23, A perturbed proportion is an expression for the case when, therp being three magnitudes a, b, c and three others A, B, C, a is to # as .^ is to C, and ^ is to f as j4 is to if. Another description of this case is found in Archimedes, "the ratios being dissimilarly ordered " (nVo^oitut TtTayji-iyiay ruy koymy). The full description of the in/ereiue in this case (as proved in v. 23), namely that a is to f as .f4 is to C, is ex aequali in perturbed proportion (8c' lO-ou iv Tviapayp,ivj} dvaXoyl^), Archimedes sometimes omits the &' Eo-ou, first giving the two profwrtions and proceeding thus: "therefore, the proportions being dissimilarly ordered, a has to e the same ratio as A has to C." The fact that Def. 18 describes a particular case in which the inference St' to-oii will be proved true seems to have suggested to some one after Theon's time the interpolation of another definition between 17 and 18 eo v.DEF. »8] NOTES ON DEFINITIONS 17, 18 137 describe the ordinary case where the argument ex atquaii holds good. The interpolated definLtion runs thus ; "an ordered proportion (T<ray/it»T; aVoXoyw) arises when, as antecedent is to consequent, so is antecedent to consequent, and, as consequent is to something else, so is consequent to something else." This case needed no description after Def, 17 itself j and the supposed definition is never used. After the definitions of Book v. Simson supplies the following axioms. I. Equimultiples of the same or of equal magnitudes are equal to one another. 3. Those magnitudes of which the same or equal magnitudes are equimultiples are equal to one another. 3. A multiple of a greater magnitude is greater than the same multiple of a less. 4. That magnitude of which a multiple is greater than the same multiple of another is greater than that other magnitude. .I". . r." .1:. T i. . ..; jL»u'.."fi 1' l" - t.-.rli ,^>, i I .1 \ I i, I- ' • <>• BOOK V, PROPOSITIONS. Proposition If there be any number of magnitudes ■whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one, that multiple also will all be of all. Let any number of magnitudes whatever AB, CD be respectively equimultiples of any magnitudes E, F equal in multitude ; I say that, whatever multiple AB is of E, that multiple will AB, CD also be of E, F. For, since ^^ is the same multiple of E that CD is of F, as many magnitudes as there are in AB equal to E, so many also are there in CD equal to F. Let AB be divided into the magnitudes AG, GB equal to E, and CD into CH, HD equal to F ; then the multitude of the magnitudes AG, GB will be equal to the multitude of the magnitudes CH, HD. Now, since ^6^ is equal to E, and CH to F, therefore AG is equal to E, and AG, CH to E, F. For the same reason GB is equal to E, and GB, HD to E, E; therefore, as many magnitudes as there are in AB equal to £, so many also are there in AB, CD equal to E, F; V. 1, a] PROPOSITIONS i, a 139 therefore, whatever multiple AB Is of E, that multiple will AB, CD also be of E, F. Therefore etc. Q. E, n. De Morgan remarks of v, i — 6 that they are "simple propositions of concrete arithmetic, covered in language which makes them iminte)!igible to modem ears. The lirst, for instance, states no more than that len acres and Un roods make ten times as much as one acre and one rood." One aim therefore of notes on these as well as the other propc^itions of Book v. should be to make their purport clearer to the learner by setting them side by side with the same truths expressed in the much shorter and more familiar modem (algebraical) notation. In doing so, we shall express magnitudes by the first letters of the alphabet, a, b^ c etc., adopting small instead of capital letters so as to avoid confusion with Euclid's lettering ; and we shall use the small letters ot, n,p etc to represent integral numbers. Thus ma will always mean m times a or the m"' multiple of a (counting i . a as the first, i . a as the second multiple, and so on). Prop. I then asserts that, if ma, mb, mc etc. be any equimultiples of a, b, i etc, then ma*mb-^mt+ ...=m (a + i + c+...). Proposition 2, J/ a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second thai a sixth is of the fourth, the sum of the first and fifth will also be the same multiple of the second that the sum of the third and sixth is of the fourth. Let a first magnitude, AB, be the same multiple of a second, C, that a third, DE, is of a fourth, F, and let a ^ , , b g fifth, EG, also be the same multiple of the second, C, that ° ^ ^ a sixth, EH, is of the fourth °" ' ' — ' — ' ' ' F; F I say that the sum of the first and fifth, AG, will be the same multiple of the second, C, that the sum of the third and sixth, DN, is of the fourth, F, For, since AB is the same multiple of C that DE is of F, therefore, as many magnitudes as there are in AB equal to C, so many also are there in D£ equal to F. For the same reason also, as many as there are in BG equal to C, so many are there also in £// equal to F; »4» ■ , BOOK V [v. a, 3 therefore, as maay as there are in the whole AG equal to C, so many also are there in the whole DN equal to F. Therefore, whatever multiple AG is of C, that multiple also is DM of F. Therefore the sum of the first and fifth, AG, is the same multiple of the second, C, that the sum of the third and sixth, DH, is of the fourth, F. . , Therefore etc. Q, E, D. To find the corresponding formula for the result of this proposition, we may suppose rt to be the " second " magnitude and b the " fourth." If now the " first " magnitude is ma, the " third " is, by hypothesis, mb ; and, if the "fifth " magnitude is na, the "si)cth" is nf>. The proposition then asserts that ma + na is the same multiple of a that mh-^ nb'\s of *. More generally, if /a, ya... and j>i, gfi... be any further equimultiples of a, b respectively, ma + na-^-pa-yqa-^ ■■■ is the same multiple of a that mb-y fib+pb -yqb -y- ,., ts of b. This extension is stated in Simson's corollary to V. a thus ; " From this it is plain that, if any number of magnitudes AB, BG., GH be multiples of another C; and as many DE, EK, KL be the same multiples of F, each of each ; the whole of the first, viz. AH, is the same multiple of C that the whole of the last, viz. DL, is of F" The course of the proof, which separates m into its units and also n into its units, practically tells us that the multiple of a arrived at by adding the two multiples is the {« + n)th multiple ; or practically we are shown that i»«n- «(j = {w 4 n) a, or, more generally, that ma + /la +pa + ... ={»i + 11 +/ + , , ,) o. - " . Proposition 3, ■ ..»>•< , J/ a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also ex aequali the magnitudes taken will be equimultiples respectively, the one of the second and the other of the fourth. Let a first magnitude A be the same multiple of a second B that a third C is of a fourth D, and let equimultiples MF, GH be taken oi A, C; I say that £F is the same multiple of S that G/f is of D. For, since £F is the same multiple of A that G// is of C, therefore, as many magnitudes as there are in £F equal to A, so many also are there in G// equal to C. T. $] PROPOSITIONS 2, 3 t4t Let £F be divided into the magnitudes £JC, KF equal to A, and GH mXo the magnitudes GL, LH equal to C\ then the multitude of the magnitudes EK, ^/^will be equal to the multitude of the magnitudes GL, LH. A- B- E- C- D - O- And, since A is the same multiple of B that C is of /), while EK is equal to A, and GL to C therefore EK is the same multiple of B that GZ, is of D. For the same reason ,' KF is the same multiple of B that LH is of Z?. Since, then, a first magnitude EK is the same multiple of a second B that a third GL is of a fourth D, and a fifth KF is also the same multiple of the second B that a sixth LH is of the fourth D, therefore the sum of the first and fifth, EF, is also the same multiple of the second B that the sum of the third and sixth, GH, is of the fourth Z>. [v. 2] Therefore etc, Q, E. D, Heiberg remarks of the use of ex aeqiiali in the enunciation of this projK}- sition that, strictly speaking, it has no reference to the definition {17) of a ratio fx atquaU. But the uses of the expression here and in the definition are, I think, sufficiently parallel, as may be seen thus. The proposition asserts that, if na, nb are equimultiples of a, b, and if m .na, m . nh are equimultiples of na, nb, then M , na is the same multiple of a that m .nils of *. Clearly the proposi- tion can be extended by taking further equimultiples of the last equimultiples and so on ; and we can prove that p .f...t!t.nais the same multiple o( a that/ ,q...m.nb is of ^, where the series of numbers p .q...m .n is exactly the same in both expression^ ; and tx atquali (&' laov) expresses the fact that the equimultiples are at the same dhtanee from 3, # in the series na, m .na... and nb, m.nb... respectively. ^100 ^ , BOOK V 1 ; [v. 3, 4 Here again the proof breaks m into its units, and then breaks n into its units ; and we are ptacticalty shown that the multiple of a arrived at, viz. m . rta, is the multiptt: denoted by the product of the numbers m, h, Le. the (m«)th multiple, or in other words that Proposition 4. ' // a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the sanu ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order. For let a first magnitude A have to a second B the same ratio as a third C to a fourth D \ and let equimultiples E, F be taken of A, C, and G, H other, chance, equimultiples of B,D\ I say that, as E is to G, so is Flo H. , A ■ ■ ■•' B E ' ■ • Q . ■ I K 1 M- 1 1 C— — - D- F 1 ■ . ... L 1 N . 1 For let equimultiples A', Z. be taken of E, F, and other, chance, equimultiples M, JV of G, H. Since E is the same multiple of A that F is of C, and equimultiples K, L oi E, ./^have been taken, therefore K is the same multiple of A that L is of C, [v. 3] For the same reason J/ is the same multiple of B that A'' is of Z?, T. 4] PROPOSITIONS 3, 4 143 " ' And, since, as A is to B,so\% C to D, • • ' - • ' and oi A, C equimultiples K, L have been taken, and of ^, D other, chance, equimultiples M, N, therefore, if K is in excess of M, L also is in excess of N^ if it is equa], equal, and if less, less, [v. Def, 5] And K, L are equimultiples of E, F, and My TV other, chance, equimultiples of G, H \ therefore, as £" is to G, so is F to H. [v. Def. 5] Therefore etc. Q. E. D. This proposition shows that, if a, b, (, d are proportionals, then «ki is to n^ as Mf is to nd; w. -t^ : and the proof is as follows : Take pma, pmc any equimultiples of ma, mc, and qnb, qnd any equimulti- ples of «*, nd. Since a : i=^ : rf, it follows [v. Def. 5] that, according as pma ■> = < gnb, pmc-> = < qnd. But the^ and ^-equimultiples are any equimultiples; , therefore [v. Def. 5] . „ , - ma : nd = me : nd. It will be observed that Euclid's phrase for taking any equimultiples of A, C and any other equimultiples of .5, Z> is " let there be taken equimulti- ples E, F o{ A, C, and G, H other, chance, equimultiples of B, D," E, F being called \aaXK woUaTAoirui simply, and G, H o^Aa, a \^v•)^tv, EiriutK iroAAairAao-ia. And similarly, when any equimultiples (/T, L) of E, F come to be taken, and any other equimultiples {M, N) of G, H. But later on Euclid uses the same phrases about the nrtu equimultiples with reference to the original magnitudes, reciting that " there have been taken, of A^ C, equimultiples K, L and of B, D, other, chamt, equimultiples M, JV" ; whereas M, JV are not any equimultiples whatever of B, D, but are any equimultiples o( the parlicu/ar multiples {G, //) which have been taken of £, D respectively, though these tatter have been taken at random. Simson would, in the first place, add 5 trvytv in the passages where any equimultiples E, F are taken of A, C and any equimultiples A', £ are taken of E, F, because the words are "wholly necessary" and, in the second place, would leave them out where M, iVare called oAAn, a inxty, liraKii TrokJuurKaria of B, D, because it is not true that of B^ D have been taken "any equimultiples whatever (« hvyt), M, N." Simson adds: "And it is strange that neither Mr Bri^s, who did right to leave out these words in one place of Prop. 13 of this book, nor Dr Gregory, who changed them into the word ' some ' in three places, and left them out in a fourth of that same Prop, 13, did not also leave them out in this place of Prop. 4 and in the second of the two places where they occur in Prop. 1 7 of this book, in neither of which they can stand consistent with truth : And in none of all these places, even in those which they corrected in their Latin translation, have they cancelled the words « hvx'^ '" 'he Greek text, as they ought to have done. The same words S. iToj(i are found in 144 BOOK V [v. 4 four places of Prop. 1 1 or this book, in the first and last of which they are necessary, but in the second and third, though they are true, they are quite superfluous ; as they likewise are in the second of the two places in which they are found in the 12th prop, and in the like places of Prop. 21, 33 of this book; but are wanting in the last place of Prop. 13, as also in Prop, 25, Book XI," As will be seen, Sirason's emendations amount to alterations of the text so considerable as to suggest doubt whether we should be justifled in making them in the absence of MS, authority. The phrase " equimultiples of A, C and other, chance, equimultiples of £, D " recurs so constantly as to suggest that it was for Euclid a quasi-stereotyped phrase, and that it is equally genuine wherever it occurs. Is it then absolutely necessary to insert i trvxt in places where it does not occur, and to leave it out in the places where Simson holds it to be wrong ? I think the text can be defended as it stands. In the first place to say "take equimultiples of A, C" is 3. fair enough way of saying take any equimultiples whatevtr of A, C. The other difliculty is greater, but may, I think, be only due to the adoption of any whatever as the translation of a, Xrv-jif. As a matter of fact, the words only mean chance equimultiples, equimultiples which are the result of random selection. Is it not justifiable to describie the product of two chance numbers, numbers selected at random, as being a " ekance number," since it is the result of two random selections ? 1 think so, and I have translated <i cruxc accordingly as implying, in the case in question, " other equimultiples whatever they may happen to be," To this proposition Theon added the following : " Since then it was proved that, if K is in excess of M, L is also in excess of N, if it is equal, (the other is) equal, and if less, less, it is dear also that, if ^ is in excess of A!*, A'' is also in excess of Z, if it is equal, {the other is) equal, and if less, less ; and foi this reason, as C is to £, so also is HXa F. PosiSM. From this it is manifest that, if four magnitudes be proportional, they will also be proportional inversely." Simson rightly pointed out that the demonstration of what Theon intended to prove, viz. that, if E, G, F, H be proportionals, they are proportional inversely, i.e. (7 is to .£ as /T is to /^ does not in Che least depend upon this 4th proposition or the proof of it ; for, when it is said that, " if A" exceeds M, Z also exceeds N etc.," this is not proved from the fact that E, G, F, H are proportionals (which is the conclusion of Prop. 4), but from the fact that A, B, C, D are proportionals. The proposition that, if A, B, C, D are proportionals, they are also proportionals inversely is not given by Euclid, but Simson supplies the proof in his Prop. B. The fact is really obvious at once from the 5th definition of Book V. (cf, p. 127 above), and Euclid probably omitted the proposition as unnecessary. Simson added, in place of Theon's corollary, the following : " Likewise, if the first has the same ratio to the second which the third has to the fourth, then also any equimultiples whatever of the first and third have the same ratio to the second and fourth : And, in like manner, the first and the third have the same ratio to any equimultiples whatever of the second and fourth," V. 4. 5] PROPOSITIONS 4, 5 r^ The proof, of course, Tollows exactly the method of Euclid's proposition itself, with the only difference that, instead of one of the two pairs of equi- multiples, the magnitudes themselves are taken. In other words, the conclu- sion that MM is to n^ as KM' is to m^ i •■■•i is equally true *hen either #« or « is equal to unity. As De Morgan says, Simson's corollary is only necessary to those who will not admit jl/'into the list M, lAf, 3 J/' etc.; the exclusion is grammatical and nothing else. The same may be said of Simson's Prop. A to the effect that, " If the first of four magnitudes has to the secotid the same ratio which the third has to the fourth : then, if the first be greater than the second, the third is also greater than the fourth ; and if equal, equal ; if less, less." This is needless to those who believe ona A to be a proper component of the list of multiples, in spite of mullut signifying many. Proposition 5. ' 1/ a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder will also be the same multiple of the remainder that the whole is of the whole. S For let the magnitude AB ht. the same multiple of the magnitude CD that the part AE subtracted is of the part CF subtracted ; I say that the remainder EB is also the same multiple of the remainder ED that the whole AB is of the whole CD. : . , . ^— 1 1 ? 1 1 ? 10 For, whatever multiple AE is uf CF, let EB be made that multiple of CG. Then, since AE is the same multiple of CF that EB ts of GC, therefore AE is the same multiple of CF that AB is of GF, [V. .] IS But, by the assumption, AE is the same multiple of CF that AB is of CD. Therefore AB is the same multiple of each of the magni- tudes GF, CD ; therefore GF is equal to CD. » Let CF be subtracted from each ; therefore the remainder GC is equal to the remainder FD. I4« BOOK V [»-S And, since AE is the same multiple of CF that EB is of GC, and GC is equal to DF, as therefore AE is the same multiple of CF that ^5 is of FD. But, by hypothesis, AE is the same multiple of CF that -(4^ is of CD ; therefore EB is the same multiple of FD that -^^ is of CZ?, That is, the remainder EB will be the same multiple of 30 the remainder FD that the whole AB is of the whole CD. Therefore etc. Q, E. D, 10. let EB be made that muJtiple of CO, ntavn-rKinaf ytYmirw rol to EB tw rB. From this way of stating; the construction one mi^t suppose that CG was given and EB had to be found equal to a certain multiple of it. But in fact EB ia what is given and CG has to be found, i.e. CG has to be constructed as a certain ju^uttiplc of EB* This proposition correspotids to V. i, with subtraction taking the place of addition. It proves the foimula ma~mb = M{a—b). '"" ' Euclid's construction assumes that, ii AE\% any multiple of CF^ and EB is any other magnitude, a fourth straight line can be found such that EB is the same multiple of it that AE is of CF, or in other words that, given any magnitude, we can divide it into any number of equal parts. This is however not proved, even of straight lines, much less other magnitudes, until vi. 9. Peletarius had already seen this objection to the construction. The difficulty is not got over by regarding it merely as a hyfothetkal construction ; for hypothetical constructions are not in Euclid's manner. The remedy is to substitute the alternative construction given by Sim son, after Peletarius and Campanus' translation from the Arabic, which only requires us to add a magnitude to itself a certain number of times. The demonstration follows Euclid's line exactly. "Take AG the same multiple of FD that AE is of CF; therefore AE is the same multiple of C^that EG is of CD. But AE, by hypothesis, is the same multiple of CF that .^^ is of CD ; therefore EG is the same multtple of CD that A£ is of CD; wherefore ^C is equal to AJ3. Take from them the common magnitude AE ; the remainder AG is equal to the remainder EB. Wherefore, since AE is the same multiple of CFthu AG is of FD, and since AG is eqaa.1 to EB, therefore AE is the same multiple of CFtitat EB is o^ FD, But AE is the same multiple of C^that AB is of CD; therefore EB is the same multiple of FD that AB is of CD." V. 5, 6] PROPOSITIONS 5, 6 147 Euclid's proof amounts to thb. Suppose a magnitude x taken such that ■• ■' ma — mi^mx, say. Add TTii to each side, whence (by v. i) Therefore a=jf + #, or *=a — *, ' *' ' ' so that ■ mi-tni> = m{a-b). ' ' Simson's proof, on the Other hand, argues thus. Take x = m{a~ i), the same multiple of (a — i) that md is ol d. Then, by addition of mfi to both sides, we have [v. i] x + m6 = ma, or at = ma — mb. • • That ia, ma~mi = m{a — i). Proposition 6, If (wo magnitudes be equimultiples 0/ two magnitudes, and any magnitudes subtracted from them 6e equimultiples of the sam4, the remainders also are either equal to the same or equi- multiples of them. For let two magnitudes AB, CD be equimultiples of two magnitudes E, F, and let AG, CH subtracted from them be equi- a q 8 multiples of the same two E, F; ' ' ' I say that the remainders also, GB, ^ o h HD, are either equal to E, F or — *■ — 1 — >- — 1 — equimultiples of them. F — For, first, let GB be equal to ^ ; I say that HD is also equal to F. For let CK be made equal to F. Since AG\% the same multiple of E that CH is of F, while GB is equal to E and KC to F, therefore AB is the same multiple of E that KH is of F. [v. 2] But, by hypothesis, AB is the same multiple of E that CDv^QiF; therefore KH is the same multiple of F that CD is of F. Since then each of the magnitudes KH, CD is the same multiple of F, therefore KH ts equal to CD, 148 ^ BOOK V [v. 6, 1 Let C// be subtracted from each ; therefore the remainder A'C is equal to the remainder //D. But F is equal to JCC ; therefore //D is also equal to J^. Hence, if GB is equal to £, HD is also equal to F. Similarly we can prove that, even if GB be a multiple of B, HD is also the same multiple of F. Therefore etc Q. E, D. This proposition corresponds to v. 3, with subtraction taking the place of addition. It asserts namel)' that, if n \& less than m, ma — na is the same multiple of a that mb-nb'\% of b. The enunciation distinguishes the cases in which m-» is equal to i and greater than i respectively. Simson observes that, while only the first case ^the simpler one) is proived in the Greek, both are given in the Latin translation from the Arabic ; and he supplies accordingly the proof of the second case, which Euclid leaves to the reader. The fact is that it is exactly the same as the other except that, in the construction, CK is made the same multiple of /"that GB is of E, and at the end, when it has been proved that KC is equal to HD, instead of concluding that'/fZ> is equal to F, we have to say " Because GB is the same multiple of E that KC is of F, and KC is equal to HD, therefore HD is the same multiple of ./^that GB is of E." Proposition 7, i. .. - > Equal magnitudes have to (he same the same ratw, as also has the same io equal magnitudes. Let A, B be equal magnitudes and C any other, chance, magnitude ; I say that each of the magnitudes A, B has the same ratio to C, and C has the same ratio to each of the magnitudes A,B. A D^ B Er. Ci < f- For let equimultiples D, E o^ A, B be taken, and of C another, chance, multiple F. Then, since D is the same multiple of A that E is of B, while A is equal to ^, therefore D is equal to E. But F is another, chance, magnitude. V. 7. 8] PROPOSITIONS 6—8 149 If therefore D is in excess of F, E is also in excess of F, if equal to it, equal ; and, if Jess, less. And D, E are equimultiples oi A, B, while F'ls another, chance, multiple of C; therefore, as A is to C, so is B to C, [v. Def. 5] I say next that C also has the same ratio to each of the magnitudes A, B. For, with the same construction, we can prove similarly that D is equal X,o E\ and F is some other magnitude. If therefore F\%\n excess of D, it is also in excess of E, if equal, equal ; and, if less, less. And /^ is a multiple of C, while D, E are other, chance, equimultiples of ^, B ; therefore, as C is to -r^ , so is C to B. [v. Def. 5] Therefore etc. PoRiSM, From this it is manifest that, if any magnitudes are proportional, they will also be proportional inversely. Q. E. D. In this proposition there is a similar use of t irvx^y to that which has been discussed under Prop. 4. Any multiple F <A C is taken and then, four lines lower down, we are told that " F is another, chance, magnitude." It is of course not any magnitude whatever, and Simson leaves out the sentence, but this time without calling attention to it. Of the Porism to this proposition Heiberg says that it is properly put here in the best ms.j for, as August had already observed, if it was in its right place where Theon put it (at the end of v. 4), the second part of the proof of this proposition would be unnecessary. But the truth is that the Porism is no more in place here. The most that the proposition proves is that, if A, B are equal, and Cany other magnitude, then two conclusions are simultaneously established, (1) that A is to C s.% B is to Cand (2) that C\%io A &.& C is to B. The second conclusion is not established from the first conclusion (as it ought to be in order to justify the inference in the Porism), but from a hypothesis on which the first conclusion itself depends ; and moreover it is not a proportion in its genera! form, i.e. between four magnitudes, that is in question, but only the particular case in which the consequents are equal. Aristotle tacitly assumes inversion (combined with the solution of the problem of Eucl. vi. 11) in Meteoroiogiea ni. 5, 37G a 14 — 16. Proposition 8. Of unequal magnitudes, the greater has to the same a greater ratio than the /ess has ; and the same has to the less a greater ratio than it has to the greater. '. ^ t» * ■< h - l-T..' L , ,. 1 N tgp 9- BOOK V fr.» -" Let AB, C be unequal magnitudes, and let AB be greater ; let D be another, chance, - n magnitude ; r say that AB has to Z? a greater ratio than C has to £), and /? has to C a greater ratio than it has to AB. For, since ^^ is greater than C, let BB be made equal toC; then the less of the magni- tudes A£, JSB, if multiphed, will sometime be greater than £f. [v, Def. 4] ICase I.] First, let A£ be less than BB ; let AB he multiplied, and let BG be a multiple of it which is greater than D ; then, whatever multiple BG is of AB, let G/f be made the same multiple of BB and ^ of C ; and let Z be taken double of D, M triple of it, and successive multiples increasing by one, until what is taken is a multiple of D and the first that is greater than K, Let it be taken, and let it be N which is quadruple of D and the first multiple of it that is greater than a. Then, since K is less than N first, therefore K is not less than M. And, since BG is the same multiple of AB that G/f is of BB, therefore BG is the same multiple of AB that B/f is of AB. [V. ,] But BG is the same multiple of AB that A' is of C ; therefore F/f is the same multiple of AB that A' is of C ; therefore B//, K are equimultiples of AB, C. Again, since GH is the same multiple of EB that K is of C and EB is equal to C, therefore GH is equal to A'. i/^ti PROPOSITION 8 f$i But K is not less than M; ■ -^ s ■ therefore neither is (7/^ less than j)/. .1 .• • • 1 And FG is greater than Z? ; therefore the whole J^ff is greater than Z>, M together. But D, M together are equal to A', inasmuch as M is triple of D, and M, D together are quadruple of D, while A'' is also quadruple of D ; whence M, D together are equal \o N. But FH is greater than M, D ; therefore /^^ is in excess of A^, ■ "-•-»"• while K is not in excess of N. ' And FH, K are equimultiples of AB, C, while N is another, chance, multiple of D ; therefore AB has to D a. greater ratio than C has to D. [v. Def. 7] I say next, that D also has to C a greater ratio than D has to AB. For, with the same construction, we can prove similarly that TV is in excess of K, while N is not in excess of FH, And A^ is a multiple of/?, while FH, K are other, chance, equimultiples of AB, C ; therefore D has to C a greater ratio than D has to AB. [v, Def. 7] \Case 2.] Again, let AE be greater than EB. Then the less, EB, if multiplied, will sometime be greater than D, \y. Def. 4] Let it be multiplied, and e q let GH be a multiple of EB * ' and greater than D ; ^ ' ^ ^ and, whatever multiple GH is ' ' ' ' of EB, let FG be made the "^ ^ — ' same multiple of AE, and K ^ ' _' • •■ of C L ■ ^ ..^ Then we can prove simi- " ' ' ' larly that FH, K are equi- "^ ' ' ' • multiples of AB, C\ and, similarly, let N be taken a multiple of D but the first that is greater than FG, so that FG is again not less than M. IS> BOOK V [v. 8 But GH is greater than D ; therefore the whole FH is in excess of D, M, that is, of N. Now A' is not in excess of N', inasmuch as FG also, which is greater than GH, that is, than K, is not in excess of N. And in the same manner, by following the above argu- ment, we complete the demonstration. Therefore etc. Q. E. I). The two separate cases found in the Greek text of the demonstration can practically be compressed into one. Also the expositor of the two cases makes them differ more than they need. It is necessary in each case to select the smaller of the two segments AE, EB of AB with a view to taking a multiple of it which is greater than D ; in the first case therefore A£ is taken, in the second EB. But, while in the first case successive multiples of D are taken in order to find the first multiple that is greater than (7^ (or A"), In the second case the multiple is taken which is the first that is greater than EG. This difference is not necessary; the first multiple of Z) that is greater than G/f would equally serve in the second case. Lastly, the use of the magnitude /C might have been dispensed with in both cases ; it is of no practical use and only lengthens the proofs. For these reasons Simson considers that Theon, or some other unskilful editor, has vitiated the proposition. This however seems an unsafe assumption ; for, while it was not the habit of the great C J reek geometers to discuss separately a number of different cases (eg, in i, 7 and t. 35 Euclid proves one case and leaves the others to the reader), there are many exceptions to prove the rule, e.g. Eucl, III. 15 and 33 ; and we know that many fundamental propositions, after- wards proved generally, were first discovered in relation to particular cases and then generalised, so that Book v., presenting a comparatively new theory, might fairly be expected to exhibit more instances than the earlier books do of unnecessary subdivision. The use of the JC is no more con- clusive against the genuineness of the proofs. Nevertheless Simson 's version of the proof Is certaimy snorter, and more- over it takes account of the case in which AE is efua/ to £B, and of the case in which AE, EB are both greater than D (though these cases are scarcely worth separate mention). " If the magnitude which is not the greater of the two AE, EB be (i) not less than D, take FG, G/f the doubles of AE, EB. But if that which is not the greater of the two AE, EB be (2) less than £>, this magnitude can be multiplied so as to become greater than £> whether it he AE or EB, Let it be multiplied until it becomes greater than D, and let the other be multiplied as often ; let EG be the multiple thus taken of AE and GJ/ the same multiple of EB , therefore EG and G/f are each of them greater than D, And, in every one of the cases, take Z the double of D, M its triple and so on, till the multiple of Z) be that which first becomes greater than GH. Let N be that multiple of D which is first greater than ff/^ and j^the multiple of D which is next less than N. V. 8, 9] PROPOSITIONS 8, 9 133' Then, because iV is the multiple of J) which is the first that becomes greater than GJf, ■ ■ ■:. . the next preceding tnultipU is not greater than G/f; that is, Gff is not less than M. And, since FG is the same multiple of AE that G/f is of EB, GH'is ttie same multiple of EB that FH moi AB\ [v. 1] wherefore FH, C/f are equimultiples of AB, EB. And it was shown that Glf^as not less than Af; and, by the construction, FG is greater than D\ therefore the whole FH\s greater than M, D together. ' , But M, D together are etjual to N ; therefore FH\% greater than N. , But Gff'ss not greater than N; and FH, GHa-tn equimultiples of AB, BE, and jVis a multiple of D\ therefore AB has to /? a greater ratio than BE (or C) has to D. [v. Def, 7] Also D has to BE a greater ratio than it has to A/i. For, having made the same construction, it may be shown, in like manner, that N is greater than GH but that it is not greater than FH; and TV is a multiple of D, 1 .^ ; . and GH, FH zxa equimultiples of EB, AB; Therefore Dhas,to EB a greater ratio than it has to AB." [v, Def. 7] The proof may perhaps be more readily grasped in the more symbolical form thus. Take the w;th equimultiples of C, and of the excess of AB over C (that is, oi AE), such that each is greater than D\ and, of the multiples of/?, let ^i? be the first that is greater than mC, and nD the ne<;t less multiple of D. Then, since wC is not less than nD, • • * ' and, by the construction, m{AE) is greater than D, the sum of wCand rii(AE) is greater than the sum of aD and D. That is, m(AB) is greater than/Z>. ... And, by the construction, mC is less than pD. Therefore [v. Def. j] AB has to D a. greater ratio than C has to D. Again, since //J is less than m(AB), i.nApD is greater than mC, <'- ■ < •' "> ' D has to C a greater ratio than D has to AB. ' ' '' ■■'■■' • *' '. Proposition 9. Magnitudes which have the same ratio to the same are equal to one another ,■ and magnitudes to which the same has the same ratio are equal. tU BOOK V [v. 9 For let each of the magnitudes ^, B have the same ratio to C ; I say that A is equal to B. For, otherwise, each of the magnitudes A, B would not ° have had the same ratio to C\ [v. 8] but it has ; therefore A is equal to B. Again, let C have the same ratio to each of the magni- tudes A, B ; I say that A is equal to B. F'or, otherwise, C would not have had the same ratio to each of the magnitudes A, B \ \y-^\ but it has ; therefore A is equal to B. Therefore etc. Q. E. D. If ii^ is to C as B is to C, or if C is to V* as C is to B, then A is equal to B. Simson gives a more expticit proof of this proposition which has the advantage of referring back to the fundamental sth and 7th definitions, instead of quoting the results of previous projxjsitions, which, as will be seen from the next note, may be, in the circumstances, unsafe. " Let A, B have each of them the same ratio to C\ A is equal to B. For, if they are not equal, one of them is greater than the other ; let A be the greater. Then, by what was shown in the preceding proposition, there are some equimultiples of A and B, and some multiple of C, such that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than that of C. Let such multiples be taken, and let ZJ, j£ be the equimultiples of A, B, and F the multiple of C, so that D may be greater than F, and E not greater than F. But, because vf is to C as ^ is to C, and of A, B are taken equimultiples D, £, and of C is taken a multiple F, and I> is greater than F, E must also be greater than F. [v. Def. s] But £ is not greater than F: which is impossible. Next, let C have the same ratio to each of the magnitudes A and S ; A is equal to B, For, if not, one of them is greater than the other ; let A be the greater. V. 9, lo] PROPOSITIONS 9, 10 iJS Therefore, as was shown in Prop. 8, there is some multiple F of C, and some equimultiples E and D al B and A, such that F is greater than E and not greater than D. But, because C is to -^ as C is to A, and /"the multiple of the first is greater than E the multiple of the second, ^the multiple of the third is greater than D the multiple of the fourth. [v, Def, 5] But ^ is not greater than D : which is impossible. Therefore A is equal to B." , , , ^ Proposition id. ,.. .^ ,, 0/ magnitudes which have a ratio to (fie same, that which has a greater ratio is greater ; and that to which the same has a greater ratio is less. For let A have to C a greater ratio than B has to C ; I say that A is greater than B. For, if not, A is either equal to B or less, Now /4 is not equal to B\ for in that case each of the magnitudes A, B would have had the same ratio to C ; [v. 7] but they have not ; therefore A is not equal to B. Nor again is A less than B ; for in that case A would have had to C a less ratio than B has to C ; [v. 8] but it has not ; therefore A is not less than B. But it was proved not to be equal either ; therefore A is greater than B. Again, let C have to ^ a greater ratio than C has to A ; I say that B is less than A. For, if not, it is either equal or greater. ' ' Now B is not equal to A ; for in that case C would have had the same ratio to each of the magnitudes A, B ; [v. 7] but it has not ; therefore A is not equal to B. tSU BOOK V [v. lo Nor again is ^ greater than W ; ' ' *■ • -> > for in that case C would have had to ^ a less ratio than it has to A ; [v. 8] but it has not ; ' •' •'= therefore B is not greater than A. But it was proved that it is not equal either ; therefore B is less than A. Therefore etc. Q. E. D- No better example can, I think, be found of the acuteness which Simson brought to bear in his critical examination of the £&mcnfs, and of his great services to the study of Euclid, than is furnished by the admirable note on this proposition where he points out a serious flaw in the proof as given in the text. For the Rrst time Euclid is arguing about greater and ieis ratios, and it will be found by an examination of the steps of the proof that he assumes more with regard to the meaning of the terms than he is entitled to assume, having regard to the fact that the definition of greater ratio (Def, 7) is all that, as yet, he has to go upon. That we cannot argue, at present, about greaifr and less as applied to mtwi m the same way as about the same terms in relation to nrngniludes is indeed sufficiently indicated by the fact that Euclid does not assume for ratios what is in Book i. an axiom, viz. th;it things which are equal to the same thing are equal to one another ; on the contrary, he proves, in Prop. 11, that ratios which are the same with the same ratio are the same with one another. Let us now examine the steps of the proof in the text. First we are told that "j4 is greater than B. For, if not, it is either equal to B or less than it. Now jJ is not equal to B ; for in that case each of the two magnitudes A, B would have had the same ratio to C: [v. 7] but they have not : therefore A is not equal to B" As Simson remarks, the force of this reasoning is as follows. If A has to C the same ratio as B has to C, then — ^supposing any equimultiples of A, B to be taken and any multiple of C— by Def. 5, if the multiple of /i be greater than the multiple of C, the multiple of B is also greater than that of C. But it follows from the hypothesis (that ^ has a greater ratio to C than B has to C) that, by Def. 7, there must be some equimultiples of A, B anA somt multiple of C such that the multiple of ^ is greater than the multiple of C, but the multiple of B is not greater than the same multiple of C. And this directly contradicts the preceding deduction from the supposition that A has to C the same ratio as B has to C ; therefore that supposition is impossible. V. lo] PROPOSITION lo ij*f The proof now goes on thus : *' Nor again is A less than B ; for, in (hat case, A would have had to C a less ratio than B has to C\ '^ but tt has not ; therefore A is not less than B." It is here that the difficulty arises. As before, we must use Def. 7. "A would have had to C a less ratio than B has to C," or the equivalent state- ment that B would have had to C a greater ratio than A has to C, means that there would have been same equimultiples of B, A and some multiple of C such that (i) the multiple of B k greater than the multiple of C, but (z) the multiple of .1^ is nat greater than the multiple of C, and it ought to have been proved that this can never happen if the hypothesis of the proposition is true, vh. that A has to C a greater ratio than B has to C: that is, it should have been proved that, in the latter case, the multiple of A is always greater than the multiple of C whenever the multiple of B is greater than the multiple of C (for, when this is demonstrated, it will be evident that B cannot have a greater ratio to C than A has to C). But this is not proved (cf. the remark of De Morgan quoted in the note on v, Def 7, p. 130), and hence it is not proved that the above inference from the supi>osi- tion that A is less than B is inconsistent with the hypothesis in the enunciation. The proof therefore fails. Sim son suggests that the proof is not Euclid's, but the work of some one who apparently "has been deceived in applying what is manifest, when understood of magnitudes, unto ratios, viz. that a magnitude cannot be both greater and less than another," The proof substituted by Simson is satisfactory and simple. . , i , r "Let A have to Ca greater ratio than B has to C; A is greater than B. For, because A has a greater ratio to C than B has to C, there are some equimultiples of A, B and some multiple of C such that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than it. [v. Def. 7] Let them be taken, and let D, E \y& equimultiples of A, B, and F a multiple of C, such that ^ , D\% greater than F^ bilt ' £ is not greater than F. « Therefore D is greater than E. And, because D and E are equimultiples of A and B, and D is greater than E, therefore A is greater than B. [Simson's 4th Ax.] Next, let C have a greater ratio to B than it has \a A\ B is less than A. For there is some multiple F of C and some equimultiples E and D ai B and A such that ^is greater than E but not greater than D. [v. Def. 7] Therefore E is less than D ; and, because E and D are equimultiples of B and A, :■ -i -m ' therefore B is less than A." IS* BOOK V [v. II ,. . _ -t; M . _ . Proposition i i. " " Ratios which are the same with the same ratio are also the same with one another. For, as W is to B, so let C be to jD, and, as C is to D, so let ^ be to A; ' I say that, as A is to B, so \% E Xo F. , .'i A- B- o- L- c E D F' ,, M N- For of Ay C, E let equimultiples G, H, K be taken, and oi B, D, Mother, chance, equimultiples L, M, N. Then since, as ^4 is to B, so is C to D, and of y4, C equimultiples G, /Thave been taken, and of B, D other, chance, equimultiples L, M, therefore, if C^ is in excess of Z, H \s also in excess of M, if equal, equal, and if less, less. Again, since, as C is to D, so is E to E, and of C, E equimultiples H, K have been taken, and of D, F other, chance, equimultiples M, N, therefore, if H is in excess of M, K is also in excess of N, if equal, equal, and if less, less. •. ^ ■ ^ -i But we saw that, if H was in excess of M, G was also in excess of Z- ; if equal, equal ; and if less, less ; so that, in addition, if G is in excess of Z, K is also in excess kAN, if equal, equal, „ , , ,. ,., „ ,, . ,^,^ ^ . and if less, less. And G, K are equimultiples oi A, £, while L, N are other, chance, equimultiples of B, F; therefore, as A is to B, so is E to E. -i Therefore etc, ' V. II, i»] PROPOSITIONS II, I » i$9 Algebraically, if - a -.b-c: d, ' .« atid c\d=t\f, then a:b = e\f. The idiomatic use of the imperfect in quoting a result previously obtained is noteworthy. Instead of saying " But it was proved that, if H is in excess of M, G is also in excess of L," the Greek text has "But if H was in excess of M, G was also in excess of L," oXAa tl vittfulyft to © rov M, hvipaxt noi TO H ToZ A. This proposition is tacitly used in combination with V. i6 and v. 14 in the geometrical passage in Aristotle, Miteorohgica 111. 5, 376 a a 2 — *6, j Proposition 12. If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents. Let any number of magnitudes A, B, C, D, E, F be proportional, so that, as A is to B, so is C xo D and E to F\ I say that, as A is to B, so are A, C, E to B, D, F. fi. B- — C e - F M- K For o{ A, C, E let equimultiples G, N, K be taken, and q{ B, D, i^ other, chance, equimultiples L, M, N. Then since, as A is to B, so is C to D, and E to F, and of A, C, E equimultiples G, H, K have been taken, and of B, D, F other, chance, equimultiples L, M, N, therefore, if G is in excess of Z, /^ is also in excess of M, and^ofiV, if equal, equal, and if less, less ; so that, in addition, if G is in excess of L, then G^ H, K are in excess of Z,, M, N, if equal, equal, and if less, less, • i ' - ^ i6o ,. BOOK V [v. 13, 13 Now G and G, H, K are equimultiples of A and A, C, E, since, if any number of magnitudes whatever are respec- tively equimultiples of any magnitudes equal in multitude, whatever multiple one of the magnitudes is of one, that multiple also will all be of all. [v. i] For the same reason L and L, M, N are also equimultiples of B and B, D, F\ therefore, as A is to B, so are A, C, E lo B, D, P. [v. Def. s] Therefore etc. Q. E. D. Algebraically, \i a : a' = b : b' = e : / eic, each ratio is equal lo the ratio {a + 6 + e+ ...) : (a' +^'4-^+ ...). This theorem is quoted hy Aristotle, El A, Nk. v. 7, 1 131 b 14, in the shortened form "the whole is to the whole what each part is to each part (respectively)." Proposition 13. If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourik a greater ratio than a fifth has to a sixth, the first will also have to the second a greater ratio than the fifth to (he sixth. For let a first magnitude A have to a second B the same ratio as a third C has to a fourth D, and let the third C have to the fourth D a greater ratio than a fifth E has to a sixth E; I say that the first A will also have to the second B a greater ratio than the fifth E to the sixth E. f, o ■ M- B — D N E- F- L- For, since there are some equimultiples of C, E, and of D, E other, chance, equimultiples, such that the multiple of C is in excess of the multiple of D, V. 13) PROPOSITIONS ii, 13 xBi while the multiple of £ is not in excess of the multiple of P, [v. Def.. 7] let them be taken, and let G, Hhe. equimultiples of C, E, and K, L other, chance, equimultiples of D, F, so that G is in excess of K, but H is not in excess of L ; and, whatever multiple G is of C, let M be also that multiple of ^, and, whatever multiple K is of D, let N be also that multiple of^. Now, since, as ^ is to B, so is C to D, and Qi A, C equimultiples M, G have been taken, and of B, D other, chance, equimultiples A^, K, therefore, if M is in excess of N, G is also in excess of K, if equal, equal, and if less, less. , .. , . ;, li' [v. Def. 5] But 6^ is in excess of AT ; .' 1. c therefore M is also in excess of A^. ' ' But H is not in excess of Z ; ' • and M, /^are equimultiples of ^, E, ., and jV, L other, chance, equimultiples of ^, F\ therefore A has to 5 a greater ratio than E has to F. [v. Def. j] Therefore etc. Q. E. D. Algebraically, if a\h = t\dt ~* ■' • and e : d->e :/ , ' , then a \ b-rt\f. After the words " for, since "in the first line of the proof, 'Hieon added " C has to i> a greater ratio than E has to F" so that " there are some equimultiples" b^an, with him, the principal sentence. The Greek text has^ after " of D, F other, chance, equimultiples," " and the multiple of C is in excess of the multiple of D...." The meaning being " such that," I have substituted this for " and," after Simson. The following will show the method of Euclid's proof. Since e:d->e\f, there will be some equirnultiples me, nu of t, e, and some equimultiples nd, ^ oftf,/, auch that momt, while me'^nf. i«« BOOK V [v. 13, 14 But, since a:i = c;d, (. ,:/v * •) therefore, according as ma > = <ni, mc> — < nd. And mi>»d; therefore ma -> nb, while (from above) mel^nf. Therefore a\b>t\f. Simson adds as a corollary the following : " If the first Kave a greater ratio to the second than the third has to the fourth, but the third the same ratio to the fourth which the fifth has to the sixth, it may be demonstrated in like manner that the first has a greater ratio to the second than the fifth has to the sixth." This however scarcely seems to be worth separate statement, since it only amounts to changing the order of the two parts of the hypothesis. Proposition 14. If a first magnitttde have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second will also he greater than the fourth; if equal, equal; and if less, less. For let a first magnitude A have the same ratio to a second .^ as a third C has to a fourth D\ and let A be greater than C ; I say that B Is also greater than D. A c 8 D For, since A is greater than C, and B is another, chance, magnitude, therefore A has to .5 a greater ratio than C has to B. [v, 8] But, as ^ is to B, so is C to Z* ; therefore C has also to /? a greater ratio than C has to B. [v. Jl] But that to which the same has a greater ratio is less ; [v. 10] therefore D is less than B ; so that B is greater than D. Similarly we can prove that, if .^4 be equal to C, B will also be equal to D ; and, if A be less than C, B will also be less than D. Therefore etc. T. 14, is] propositions 13—15 163 Algebraically, if a : i = t : d, then, according aso> = <f, fi> = <<f, Simson adds the specific proof of the second and third parts of this proposition, which Euclid dismisses with "Similarly we can prove...." " Secondly, if v^ be equal to C, B is equal to D\ for /4 is to 5 is C, that is A, is to Z> ; therefore B is equal to D. [v. 9] Thirdly, if ^4 be less than C, B shall be less than D, For C is greater than A ; and, because C'\iXo D »s A is to B, D is greater than B, by the first case. >, , . Wherefore B is less than D." Aristotle, Mtteorol. iii. 5, 376 a ti— i4i quotes the equivalent proposition that, if a>^, e->d. Proposition 15. Parts have the same ratio as the same multiples of them taken in corresponding order. For let AB be the same multiple of C that DE is of /"; I say that, as C is to F, so\s AB to DE. Af 1 1 tB Ct- Oi ■ ' 'E f'- For, since AB is the same multiple of C that DE is of E, as many magnitudes as there are in AB equal to C, so many are there also in DE equal to E. Let AB be divided into the magnitudes AG, GH, HB equal to C, and DE into the magnitudes DK, KL, LE equal to E \ then the multitude of the magnitudes AG, GH, HBW\\\ be equal to the multitude of the magnitudes DK, KL, LE. And, since AG GH, HB are equal to one another, and DK, KL, LE are also equal to one another, therefore, as AG is to DK, so is GH to KL, and HB to LE. {^- 7] Therefore, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents ; [v. la] therefore, as ^G is to DK, so is AB to DE. i«4 -- BOOK V •"• [v. IS, '6 But AG is equal to C and DK to F; therefore, as C is to i^ so is AS to I^S, Therefore etc. v,, , q, e. d. Algebraically, a : b~ma : mi. Proposition i6. If four magnitudes be proportional, they will also be proportional alternately. Let A, B, C, D )x. four proportional magnitudes, so that, as A is to ^, so is C to Z? ; I say that they will also be so alternately, that is, as W is to C, so is B to D. A c o- E< 1 1 1 Qi 1 Fi 1 1^ 1 Hi '-—I For o{ A, B let equimultiples E, F be taken, ,, ,i . and of C, D other, chance, equimultiples G, H. Then, since E is the same multiple of A that F is of B, and parts have the same ratio as the same multiples of them, [v. is] therefore, as ^ is to ^, so is £* to F. But as ^ is to 5, so is C to /? ; therefore also, as C is to D, so is E to F. [v. ii] Ag^in, since G, H are equimultiples of C, D, therefore, as C is to /?, so is C to H. [v. ij] But, as C is to Z?, so is ^ to F\ therefore also, as ^ is to F, so is G to H. [v. n] But, if four magnitudes be proportional, and the first be greater than the third, the second will also be greater than the fourth ; if equal, equal ; and if less, less. •. ,-v-y^,- [v- m] Therefore, if E Is in excess of G, F is also in excess of H, if equal, equal, and if less, less. V. i6] PROPOSITIONS 15, i6 1C5 Now E, 7^ are equimultiples oi A, B, and G, H other, chance, equimultiples o{ C, D\ therefore, as A is to C, so is B to D. [v. Def. s] Therefore etc. Q, E. D. 3, " Let A, B, C, D be four proportignsl magnitudes, so that, as A Is to B, so fs C to D." In a number of expressions like this it is absolutely necefisffty, wheit translatitig inta Endbh, to interpolate words which are not in the Greek. Thus the Greek here is ; litfTup Tiaaapa liiP^&Ti d^^Xtyyoi' r^ A, U, T, A, i^t rh A rp6% TO B, oih-iifj ti r rpii tA A, literally *' Let At B^ C, D he four proportional magnitudes, as ^ to ^, 60 C to Z>^" The same remark applies to the eotresponding expressions in the neil proposiiions, v. 17, 18, and to other forms of expression in V. to — 13 and later propositions : e^g. in v, 10 we have a phrase meaning literally '*I^t there be mngnitudes... which taken two and two are in the same ratio, as if to ^, so Z> to £," etc.: in v. it " (magnitudes)... which taken two and two are in the same r^tio, atvd ht the proportion of them be perturbed, as ^ to ^, so £ to /",'' etc. In all such cases (where the Greek is so terse as to be almost ungrammatical) I shall insert the words necessary in English, without further remark. Algebraically, if a : b^c : d, then a; ( = b : d. Taking equimultiples /fta, mb of a, b, and equimultiples m, nd of (,d, we have, by v. 15, a : b = Ma : mb, c ; d=ne : nd. And, since a : b = e ; d, we have [v. 11] ma : mb = n£ md. Therefore [v. 14], according as wa > = < «<r, mb> = <ii4,' so that a : e = b : d. Aristotle tacitly uses the theorem in MetetfrologUa ill. 5, 376 a xz — 34. The four magnitudes in this proposition mtist all be ^ the same kind, and Simson inserts " of the same kind " in the enunciation. This is the first of the propositions of Eucl. v. which Smith and Bryant {Euclid's Ekmtnti of Geomttry, tgoi, pp. 298 sqq.) prove by means of vi, i so far as the only geometrical magnitudes in question are straight lines or rectilineal areas \ and certainly the proofs are more easy to follow than Euclid's. The proof of this proposition is as follows. To prove that, If Jour magnitudes of the same kind [straight lines or rectilineal areas] be proportionals, they will be proportionals -when taken alttmately. Let F, Q, /{, Sbe the four magnitudes of the same kind such that P:Q = R:S; then it is required to prove that '. ■^ . 1 < ; P-.R^Q-.S. \ -' First, let all the magnitudes be areas. Construct a rectangle abed equal to the area P, and to be apply the rectangle beef equal to Q, Also to cd>, bf apply rectangles ag, bk equal to JF, S respectively. ' ' ■ m BOOK V [v. i6, ij Then, since the rectangles ac, be have the same height, they are to one another as their bases, [vl. i] Hence P:Q = ah:bf. But P:Q = R:S. Therefore R •S = ab:b/y [v. 1 1] i.e. rect. ag : rect. At = ab : bf. Hence (by the converse of vi. i) the rect- angles ag, bk have the same height, so that k is on the line kg. Hence the rectangles ae, ag have the same height, namely ab ; also ^, bk have the same height, namely h/. Therefore rect. ac • rect ag=bc \ and rei;t be : recL bk = bc ; bg. Therefore rect, ac : rect, ag - recL be ; rect, bk. That is, P:Ji=Q:S. Se(ondl}\ let the magnitudes be straight lines AB, BC, CD, DE. Construct the rectangles Ab, Be, Cd, Dt with the same height. i t t a f b h 3 k bg. [VI- [v. Il] a be d e A f i i J 1 3 E Then Ab\Bc = AB\BC, aiid Cd : De= CD : DE. But AB:BC=CD:DE. Therefore Ab .Bc=Cd: Dt. Hence, by the first case, Ab: Cd=Be-De, and, since these rectangles have the same height, AB: CD = BC : DE. [VI. i] [V. „] Proposition 17. 1/ magnitudes be proportional componendo, they will also be proportional separando. Let AB, BE, CD, DF be magnitudes proportional com- ponendo, so that, as AB is to BE, so is CD to DF\ I say that they will also be proportional separando, that is, as AE is to EB, so is CF to DF. For of AE, EB, CF, FD let equimultiples GH, HK, LM, MN be taken, and of EB, FD other, chance, equimultiples, KO, NP, V. 17] PROPOSITIONS i6, ij t6? Then, since GH is the same multiple of AE that HK is oiEB, therefore GH is the same multiple of AE that GK is of AB. [V. ,] But GH is the same multiple of AE that LM is of CF\ therefore GK is the same multiple of AB that LM is of C/". "E — B e — r~B H K O Again, since LM is the same multiple of CF that MN is of FD, therefore LM is the same multiple of CF that LN is of CD. Iv. l] But LM was the same multiple of CF that CA' is of AB \ therefore GK is the same multiple of AB that LN is of CD. Therefore GK^ LN 3.r& equimultiples oi AB, CD. Again, since HK is the same multiple of EB that MN is of/Z?, and KO is also the same multiple of EB that NP is of /"/>, therefore the sum HO is also the same multiple of EB that MP is of /^£'. [v. i] And, since, as AB is to ^^, so is CD to /?/% and oi AB, CD equimultiples GK, LN have been taken, and of EB, FD equimultiples HO, MP, therefore, if GK is in excess of HO, LN is also in excess of MP, if equal, equal, , . i, and if less, less. Let GK be in excess of HO ; then, if HK be subtracted from each, ' GH is also in excess of KO. But we saw that, if GK was in excess of HO, LN was also in excess of MP ; therefore LN is also in excess of MP, i ag^ BOOK V [v. 17 and, if MN be subtracted from each, LM is also in excess of NP ; so that, if GH is in excess of KO, LM is also in excess of NP. Similarly we can prove that, if GH be equal to KO, LM will also be equal to NP^ and if less, less. And GH, LM are equimultiples of AE, CF, while KO, NP are other, chance, equimultiples of EB, FD ; therefore, as AE is to EB, so is CF to FD. Therefore etc. Q. E. D. Algebraically, if a \b = c ; d, then {a-b):b = (c-d)\d. I have already noted the somewhat strange use of the participles of (TuyKttcr^cu and Stai/xur^at to convey the sense of the technical ^^ivB^a^^ and WiptiTK Xoyou, or what we denote by (ompontndo and sefarafido. lax <rvyi«iV«'a fiiyiOr) dirdkoyov J, itai StaifitOivra draXjryov IcrtOi is, literally, "if magnitudes compounded be proportional, they will also be proportional separated," by which is meant "if one magnitude made up of two parts is to one of its parts as another magnitude made up of two parts is to one of its parts, the remainder of the first whole is to the part of it first taken as the remainder of the second whole is to the part of it first taken." In the algebraical formula above a, c are the wholes and b,a-b and d, c-^are the parts and remainders respectively. The formula might also be stated thus ; If a-^b •.b=-c ^d vd, then a : b = c : d, in which case a + #, c + d are the wholes and 0, a and d, i the parts and remainders respectively. Looking at the last formula, we observe that "separated," Siatp«fl(fTa, is used with reference not to the magnitudes a, i, c, d but to the comfamndid magnitudes a + b, b, c + d, d. As the proof is somewhat long, it will be useful to give a conspectus of it in the more symbolical form. To avoid minuses, we will takf for the hypothesis the form a + b\s xa b zs c + d\sUi d. Take any equimultiples of the four magnitudes a, b, t, d, viz. ma, mb, ntt, md, and any other equimultiples of the consequents, viz. nb and nd. Then, by v. i, m{a + b), ni {c + d) are equimultiples of a + i, c-¥d, and, by v. 2, (m + «) b, {tn + n)d are equimultiples of b, d. Therefore, by Def, s, since a + b is to 6 its c+d is to d, according as (w (« + i) > = <{« + /«) #, ot {f + <()>-< (m + «) rf. V. I J, i8] PROPOSITIONS t;, i8 z^ Subtract from m (a + i), (m + n)i the common part mi, and from m(c + d), {m + »)d the common part md; and we Itave, according as ma> = <n6, mc> = <nd. But ma, mc are any equimultiples of «, e, and nb, nd any equimultiples of kd, therefore, by v. Def. 5, a is to ^ as r is to d. Smith and Bryant's proof follows, mutatis mutandis, their alternative proof of the next proposition (see pp. 173 — 4 below). Proposition 18. If magnitudes be proportional separando, they will also be proportional com pone ndo. Let AE, EB, CF, FD be magnitudes proportional separando, so that, as AE is to EB, so is CF to FD ; a e b I say that they will also be ' ' ^ proportional componendo, that a ^ & is, as AB is to BE, so is .1 CD to FD. For, if CD be not to DF as AB to BE, then, as AB is to .5.5', so will CD be either to some magnitude less than DF or to a greater. First, let it be in that ratio to a less magnitude DG, Then, since, as AB is to BE, so is CD to DG, they are magnitudes proportional componendo; so that they will also be proportional separando. [v. 17] Therefore, as AE is to EB, so is CG to GD. But also, by hypothesis, as AE is to EB, so is CF to FD. Therefore also, as CG is to GD, so is CF to FD. [v, it] But the first CG is greater than the third CF; therefore the second GD is also greater than the fourth FD. [v. 14] But it is also less : which is impossible. Therefore, as AB is to BE, so is not CD to a less m^nitude than FD. Ifo BOOK V [v. i8 Similarly we can prove that neither is it in that ratio to a greater; it is therefore in that ratio to FD itself. Therefore etc. Q. E. D, Algebraically, if a \b = e : d^ then {a^b) \ b^{f-kd) ; d. In the enunciation oC this proposition there is the same special use of Siijp>jli.(ya and crvtri^iVra as there was of cruyKtiftira and SuupiSiyra in the last enunciation. Practically, as the algebraical form shows, Stgpriiiira, might have been left out. The following ts the method of proof employed by Euclid, Given tnat a:b = e:d, suppose, if possible, that (o + jS) ■.b = {c + d):{d±x). Therefore, J<^ir«i«<fo [v. 17], a : b = {c + x) •.{d±x), whence, by v. 1 1, {cT x) ; {d ±x) = c : d. But {e—x)< c, while {d + x)> d, and {c-\-x)>e, while {d-x)<d, which relations respectively contradict v. 14. Simson pointed out (as Saccheri before him .saw) that Euclid's demonstra- tion is not legitimate, because it assumes without proof that to any three magnitudes, two of which, at ieasi, are 0/ the same kind, there exists a fourth pr&poriionai. Clavius and, according to him, other editors made this an a;<iom. But it is far from axiomatic ; it is not till vi. 1 2 that Euclid shows, by construction, that it is true even in the particular case where the three given magnitudes are alt straight lines. In order to remove the defect it is necessary either (r) to prove beforehand the proposition thus assumed by Euclid or (2) to prove v. tS independently of it. Saccheri ingeniously proposed that the assumed proposition should be proved, for areas and straight Hues, by means of Euclid vi. i, 2 and 12. As he says, there was nothing to prevent Euclid from interposing these proposi- tions immediately after v. 17 and then proving v. 18 by means of them. VI. 12 enables us to construct the fourth proportional when the three given magnitudes are straight lines ; and vi. 1 2 depends only on vi. i and 2. " Now," says Saccheri, " when we have once found the means of constructing a straight line which is a fourth proportional to three given straight lines, we obviously have the solution of the general problem ' To construct a straight line which shall have to a given straight line the same ratio which two polygons have {to one another).'" For it is sufficient to transform the polygons into two triangles of equal height and then to construct a straight line which shall be a fourth proportional to the bases of the triangles and the given straight line. The method of Saccheri is, as will be seen, similar to that adopted by V. i8] PROPOSITION i8 ift Smith and Bryant {/i^. at.) in proving the theorems of Euclid v. i6, 17, 18, 21, so far as straight lines and rectilineal areas are concerned, by means of vi. i. De Morgan gives a sketch of a general proof of the assumed proposition that, B being any magnitude, and P and Q two magnitudes of the same kind, there does exist a magnitude A which is to ^ in the same ratio as /" to Q. " The right to reason upon any aliquot part of any magnitude is assumed ; though, in truth, aliquot parts obtained by continual bisection would suffice : and It is taken as previously proved that the tests of greater and of less ratio are never both presented in any one scale of relation as compared with another" (see note on v. Def, 7 ad ^n,). "(i) If ^be to ^ in a greater ratio than Pto Q, so is every magnitude greater than AT, and so are s&me leu magnitudes ; and if jW be to ^ in a less ratio than P to Q, so is every magnitude less than M, and so are some greater magnitudts. Part of this is in every system : the rest is proved thus. If j^ be to J in a greater ratio than P to Q, say, for instance, we find that isjI/ lies between »i2? and 23i?, while 15/" lies before J2Q. Let \^M exceed 22^? by Z; then, if iV be less than M by anything less than the 15th part of Z, i^N is between ziB and 23^; or JVJ less than M, is in a greater ratio to B than P to Q. And similarly for the other case, (2) ,^can certainly be taken so small as to be in a less ratio to B than P to Q, and so large as to be in a greater ; and since we can never pass from the greater ratio back again to the smaller by increasing M, it follows that, while we pass from the first designated value to the second, we come upon an intermediate magnitude A such that every smaller is in a less ratio to B than P to Q, and every greater in a greater ratio. Now A cannot be in a less ratio to B than P to Q, for then some greater magnitudes would also be in a less ratio ; nor in a greater ratio, for then some less magnitudes would be in a greater ratio; therefore A is in the same ratio to 2? as ^ to Q. The previously proved proposition above mentioned shows the three alternatives to be the only ones." Alternative proofs of V. 18. Simson bases his alternative on v, 5, 6, As the 18th proposition is the converse of the 17th, and the latter is proved by means of v. i and j, of which V. 5 and 6 are converses, the proof of v. 18 by v. 5 and 6 would be natural; and Simson holds that Euclid must have proved v, i3 in this way because "the sth and 6th do not enter into the demonstration of any proposition in this book as we have it, not can they be of any use in any proposition of the Elements," and "the sth and 6th have undoubtedly been put into the 5th book for the sake of some propositions in it, as all the other propositions about equimultiples have been." Simson's proof is however, as it seems to me, intolerably long and difficult to follow unless it be put in the symbolical form as follows. Suppose that a is to i as (^ is to rf; , ..",.'" ; it is required to prove that a 4- j is \a b »&c-¥d\&X.o i. Take any equimultiples of the last four magnitudes, say w(a + i), iiib, m{c + d), md, and any equimultiples of i, d, as nb, nd. ^fM. BOOK V [v. iS Oearly, if «d is greater than mi, ... lid is greater than md; if equal, equal ; and if less, less. I Suppose nd not greater than mi, so that nd is also not greater than W. Now m(a4-i') is greater than mi ; therefore m[a + i) is greater than »6. Similarly m (^ + rf) is greater than nd. II. Suppose ni greater than mi. Since « (a + i), md, m{e + d), md are equimultiples of (o + i), i, (f + rf)i <^> ma is the same multiple of a that m{a-y i) is of {a + i), and mc is the same multiple of c that m (f + d) is of (c + d), so that ffio, mc are equimultiples of a, c. [v. 5] Again n^, nd are equimultiples of J, d, and so are m^, md\ therefore (n-m)i, {H-m)d are equimultiples of b, d and, whether n-m is equal to unity or to any other integer [v. 6^ it follows, by Def. S, that, since a, b, r, d are profxsrtionals, if ma is greater than {n-m)i, then mc is greater than {n-m)d; if equal, equal ; and if less, less. (i) If now m{a-\-i} is greater than ni, subtracting mi from each, we have ma is greater than {n-m)i; therefore mc is greater than (n - m)d, and, if we add md to each, m(c-¥d) is greater than nd. (3) Similarly it may be proved that, if w (a + ^) is equal to ni, then j» (f + rf) is equal to nd, and (3) that, if m(aA- i) is less than tti, then mic + d) is less than nd. But (under I. above) jt was proved that, in the case where ni is not greater than mb, m{a +i) a always greater than ni, and m(c + d) is always greater than nd. Hence, whatever be the values of m and n, m {c + d) is always greater than, equal to, or less than nd according as m(a + i) is greater than, equal to, or less than nb. Therefore, by Def. 5, a+i is U>baac + d is tod, Todhunter gives the following short demonstration from Austin {Exami- nation 0/ the fin t six books of Euclid's Elements). "Let AE be to J?J? as CFis to FD: AB shall be to BE as CD is to DF. V. i8] PROPOSITION i8 ^ For, because AE is to EB as CF'xs to FD, therefore^ alternately, AE is to CFtts EB is to FD. [v. i6] And, as one of the antecedents is to its consequent, so is the sum of the antecedents to the sum of the consequents: [v. 12] therefore, as EB is to FD, so are AE, EB together to CF, FD together ; that is, AB is to CD as EB is to FD. Therefore, alternately, AB is to BE as CD is to FD." The objection to this proof is that it is only valid in the case where the proposition v. t6 used in it is valid, i.e. where all four magnitudes aie of the same kind. Smith and Bryant's proof avails where all four magnitudes are straight lines, where all four magnitudes are rectilineal areas, or where one antecedent and its consequent are straight lines and the other antecedent and its consequent rectilineal areas. .. , Suppose that A : B= C : D. First, let all the magnitudes be areas. Construct a rectangle abed equal to A, and to be apply the rectangle baf equal to B. Also to ab, bj apply the rectangles ag, bk equal to C, D respectively. Then, since the rectangles m, be have equal heights be, they are to one another as their base& [vi. i] Hence ab:bf= recL tu : rect bt = C:D = rect, ag ; rect bk. Therefore [vi. i, converse] the rectangles ag, bk have the same height, so that i is on the str^ght line hg. Hence A -^ B\B- recL ae ; rect. bt = af:bf = rect ak : rect bJi = C+D:D. SMondiy, let the magnitudes A, B'^ straight lines and the magnitudes C, i> areas. Let ab, if \x equal to the straight lines A, B, and to ab, bf apply the rectangles ag, bk equal to C, D respectively. Then, as before, the rectangles ag, bk have the same height. Now A + B:B^a/.bf = rect. aJk : rect. bk = C + D:D. i ,_ , ,. Thirdly, let all the magnitudes be straight lines. Apply to the straight lines C, D rectangles F, Q having the same height. <t < b J k t 1 k ^4 BOOK V [v. 18, 19 Then P:Q=C:D. [v.. l] Hence, by the second case, Also P+Q:Q=C+D.D. Therefore A + B:B = C+I):£>. 11 Proposition 19 -., ' If f as a whoU is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole. For, as the whole AB is to the whole CD, so let the part AE subtracted be to the part CF subtracted ; I say that the remainder EB will also be ■ ? to the remainder FD as the whole AB to c ^ d the whole CD. J For since, as AB is to CD, so is AE to CF, alternately also, as BA is to AE, so is DC to CF. [v. 16] And, since the magnitudes are proportional componendo, they will also be proportional separando, [v, 17] that is, as BE is to BA^ so is DF to CF^ and, alternately, as BE is to DF, so is EA to FC. [v, ifi] But, as AB is to CF, so by hypothesis is the whole AB to the whole CD. Therefore also the remainder EB will be to the remainder FD as the whole AB is to the whole CD. [v. 1 1] Therefore etc. [PoRisM. From this it is manifest that, if magnitudes be proportional componendo, they will also be proportional eonvertendo^ p. E. D. Algebraically, \l a:b = c\d (where e<.a and d < fi), then {a-<;):{i-d) = a:i. The " Porism " at the end of this projwsition is led up to by a few lines which Heiberg brackets because it is not Euclid's habit to explain a Porism, and indeed a Porism, from its very iwture, should not need any V. 19, so] PROPOSITIONS i8— ao it^ explanation, being a sort of by-product appearing without effort or trouble, air/nYfioTtvruf (Proclus, jpt 303, 6). But Heiberg thinks that Simson does wrong in finding fault with the argument leading to the "Porism," and that it does contain the true demonstration of conversion of a ratio. In this it appears to me that Heiberg is clearly mistaken, the supposed proof ott the basis of Prop. 19 being no more correct than the similar attempt to prove the inversion of a ratio from Prop. 4. Thu words are : " And since it was proved that, as A£ is to CB, so is EB to FJD, alternately also, as AB is to BE, so is CD to FD : therefore magnitudes when compounded are proportional. ,., But it was proved that, as £A is to AE, so is DC to CE and this is foniferiendo." It will be seen that this amounts to proving /ram the hypothesis a:b = €\d that the following transformations are simultaneously true, viz. : &\a~e-=b:h~d, and a'.c-b:d. The former is not proved from the latter as it ought to be if it were intended to prove conversion. The inevitable conclusion is that both the "Porism" and the argument leading up to it are interpolations, though no doubt made, as Heiberg says, before Theon's time. The conversion of ratios does not depend upon v. rg at all but, as Simson shows in his Proposition E {containing a proof already given by Clavius), on Props. 17 and 18. Prop. E is as follows. If four magnitudes U proportionals^ they are also proportionals by conversion, that is, tht first is to its excess above the second as the third is to its excess above tht fourth. ' Let .^J be to BE as CD to DF: then BA hloAE?ts DC to CE ^ ° Because AB is to BE as CD to DF, F by division [s^arando], AE'is to EB as CEto FD, [v. ij] and, by inversion, BE is to EA as DE to EC. [Simson's Prop, B directly obtained from v, Def. 5] Wherefore, by composition [com^nendo], BA is to AE as DC to CE [v. 18] Proposition 20. If there be three magnitudes, and others equal to them in multitttde, which taken two and two are in the same ratio, and if ^x aequali the first be greater than the third, the fourth will also be greater than the sixth; if equal, equal; and, if less, less. ff6 BOOK V [v. lo Let there be three magnitudes A, B, C, and others D, £, F equal to them in multitude, which taken two and two are in the same ratio, so that, ■ as /^ is to J3t so is D to E, •■ and as B is to C, so is ^ to i^; and let y4 be greater than C ex aequali ; I say that D will also be greater than F\ \i A is equal to C, equal ; and, if less, less. A o- B E- c— F- For, since A is greater than C, and B is some other magnitude, and the greater has to the same a greater ratio than the less has, [v. 8] therefore A has to ^ a greater ratio than C has to B. But, as ^ is to B, so is D to E, and, as C is to B, inversely, so is /^ to ^ ; therefore Z? has also to ^ a greater ratio than 7^ has to ^, [v. 13] But, of magnitudes which have a ratio to the same, that which has a greater ratio is greater ; [v. 10] therefore D is greater than F. Similarly we can prove that, if j4 be equal to C, D will also be equal to F ; and if less, less. Therefore etc. Q. E. D. Though, as already remarked, Euclid ha£ not yet given us any definition of cempoanded ratios. Props, 20 — 23 contain an important part of the theory of such ratios. The term "compounded ratio" is not used, but the propositions connect themselves with the definitions of ex atguali in its two forms, the ordinary form defined in Def. 1 7 and that called ptrturbid proportion in Def. 18. The compounded ratios dealt with in these propositions are those compounded of successive ratios in which the consequent of one is the antecedent of the next, or the antecedent of one is the consequent of the next. Prop. Z2 states the fundamental proposition about the ratio tx aequali in its ordinary form, to the effect that, if a is to ^ as ^ is to f, and ^ is to If as « is to/, then a is to ^ as ^ is to/ V. ao] PROPOSITION 20 t JJ with the extension to any number of such ratios ; Prop, 23 gives the corresponding theorem for the case al perturbed proportion, namely thati if a is to ^ as « is to ^ .,^i and j is to ^ as </ is to e, then a IS to f as rf is to/ Each depends on a preliminary proposition. Prop, a a on Prop. 20 and Prop. 33 on Prop, a i. The course of the proof will be made most clear by using the algebraic notation. The preliminary Prop, 20 asserts that, if a ■.b = d:e, and t:c-t:/, then, according as «> = <<■, ds- = <./. For, according as a is greater than, equal to, or less than c, the ratio o ; * is greater than, equal to, or less than the ratio ( : b, [v, 8 or v. 7] or (since d:e = a:b, and £:b=/:e) the ratio d:e\s greater than, equal to, or less than the ratio/; t, [by aid of V. 13 and v. 11] and therefore d is greater than, equal to, or less than/ [v. 10 or v. 9] It is next proved in Prop. 22 that, by v. 4, the given proportions can be transformed into ma : nb = md : tie, and nb : pc = ne : pf, whence, by v. 20, according as ww is greater than, equal to, or less than, pc, md is greater than, equal to, or less than^ 80 that, by Def. 5, • ■ ■ - . ' a:e = d:/. Prop, 23 depends on Prop. 21 in the same way as Prop. 22 on Prop, ao, but the transformation of the ratios in Prop. 23 is to the following : (i) ma : mb = ne : ttf (by a double application of v. 1 5 and by v. 11), (a) mb '.nc ~md\nt (by V. 4, or equivalent steps), and Prop, a I is then used. ' '" . r . ..^ Simson makes the proof of Prop, 20 slightly more explicit, but the main difference from the text is in the addition of the two other cases which Euclid dismisses with " Similarly we can prove." These cases are ; "Secondly, let A be equal to C; then shall D be equal to F. Because A and C are equal to one another, j4 is to .5 as C is to jB. [v, 7] But ■ ' ' A \% ifi B a& D '^ vo E, ■'.'., and C is to .f as .F is to E, wherefore i? is to £■ as 7^ to E ; . 1 -i [v. 11] and therefore Z> is equal to .^ ' 1 • .• [v. g] •..1.1 ifg BOOK V [v. ao, « Next, let A be less than C; then shall JJ be less than J^. For C is greater than A, ■ ' - and, as w^ shown in the lirst case, Cis to Jas i^to^, and, in like manner, J is to vSt as ^ to i? ; therefore F is greater than D, by the first case ; and therefore D is less than JK" Proposition 21. // there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then, if ex aequali the first magnitude is greater than ths third, the fourth will also be greater than the sixth ; if equal, equal; and if less, less. Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two are In the same ratio, and let the proportion of them be perturbed, so that, as A is to i?, so is ^ to F, , and, as ^ is to C, so is /? to E, and let A be greater than C ex aequali ; I say that D will also be greater than F\ if A is equal to C, equal ; and if less, less. A D- B^ — E- o F- For, since A is greater than C, and B is some other magnitude, therefore A has to ^ a greater ratio than C has to B. [v. 8] But, as A is to B, so is E to F, and, as C is to B, inversely, so is A to D. Therefore also E has to ^a greater ratio than E has to JD. [v-3] But that to which the same has a greater ratio is less ; [v. 10] therefore F is less than D ; therefore Z? is greater than F, V. 31, ijj PROPOSITIONS ao— 22 ^^ Similarly we can prove that, f if ^ be equal to C, D will also be equal to F\ and if less, less. Therefore etc. q. e. d. Algebraically, if a:b = e:f, and b:c=d\€, then, according asa> = <f, ;/> = <f. Simson's alterations correspond to those which he makes in Prop, a a. After the first case he proceeds thus. "Secondly, let A be equal to C; then shall D be equal to F. Because A and Care equal, H,i is to ^ as C is to ^. [v. 7] But /4 is to ^ as £ is to .^ and C is to .# as £ is to ^ : wherefore E is to /"as E to D, [v. 11] and therefore D is equal to F. [v. 9] Next, let A be less than C\ then shall D be less than F. For C is greater than A^ and, as was shown,* C is to .5 as .E to 2?, and, in like manner, .ff isto^as F\.aE\ therefore i^is greater than D, by the first case, and therefore D is less than F'^ The proof may be shown thus. According as (J > = < f, a;h> = <(;h. ' But a:b = e:fy and, by inversion, c:b = t:d. Therefore, according as «> = <<:, e:/> = <e:d, and therefore d> = </. Proposition 22. 1/ there be any number 0/ magniiudes whatever, arui others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio ex aequali. Let there be any number of magnitudes A, B, C, and others Z>, £, F equal to them in multitude, which taken two and two tc^ether are in the same ratio, so that, as j4 is to .5, so is Z* to E^ and, as .5 is to C, so is £" to ^; I say that they will also be in the same ratio ex aequali, < that is, as .^ is to C, so is D to F> . xStt it— BOOK V [v. 21 For of A, D let equimultiples G, H be taken, and of B, E other, chance, equimultiples A', L ; and, further, of C, F other, chance, equimultiples J/, N. A B c- D E- — F- — I K 1 ► Then, since, as A is to B, so is Z? to £", and of A, D equimultiples G, H have been taken, and of B, E other, chance, equimultiples K, L, therefore, as 6^ is to K, so is H to L. [v. 4] For the same reason also, as A' is to M, so is L to N. Since, then, there are three magnitudes G, K, M, and others H, L, N equal to them in multitude, which taken two and two together are in the same ratiQ, therefore, ex aeguali, if G is in excess of M,H\s also in excess oiN; if equal, equal; and if less, less. ■ - .i [v. ao] And G, H are equimultiples o( A, D, and M, N other, chance, equimultiples of C, F. Therefore, as ^ is to C, so is D to F. [v. Def. 5] Therefore etc, Q. E. D. EucUd enunciates this proposition as true of any number of magnilvdti whatetier forming two sets connected in the manner described, but bis proof is confined to the case where each set consists of three magnitudes only. The extension to any number of magnitudes is, however, easy, as shown by Simson. "Next let there be four magnitudes A,B,C, D, and other four E, F, G, Jf, which two and two have the same ratio, viz. : as /4 is to ^, so is -£ to ./% A B C D E F O H and as £ is to C, so is .^to G, and as C is to A so is C to .ff ; A shall he to D as £ to IT. Because A, S, C are three magnitudes, and E, .f, G other three, which taken two and two have the same ratio, by the foregoing case, v^ is to C as £ to 6^. y. 31, *3) PROPOSITIONS ai, aj But C is to Z) as C is to If; wherefore again, by the fiist case, ^ is to Z> as £ to /^ And so on, whatever be the number of magnitudes." Proposition 23. // there be three magnitudes, and others equal to them in mtdtilude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, they will also be in the same ratio ex aequali. Let there be three magnitudes A, B, C, and others equal to them in multitude, which, taken two and two together, are in the same proportion, namely D, E, F\ and let the propor- tion of them be perturbed, so that, as >4 is to B, so is £" to F, and, as j9 is to C, so is Z? to ^ ; ' 1 say that, as ^ is to C, so is Z? to .F. A B — c D E^ F- — - O 1 1 H 1 1 L ■ K 1 1 M 1 N' 1 Of W, B, D let equimultiples G, H, Kh^ taken, and of C, E, Z^ other, chance, equimultiples L, M, N. Then, since G, //^are equimultiples oi A, B, and parts have the same ratio as the same multiples of them, [v. is] therefore, as A is to .5, so is G^ to H. For the same reason also, as .£■ is to /% so is i?/ to A^. And, as .^ is to B, so is E to E\ therefore also, as G is to H, so is M to N. [v. 11] Next, since, as ^ is to C, so is D to E, alternately, also, as B is to D, so is C to E. [v. i«] And, since H, K are equimultiples of B, D, and parts have the same ratio as their equimultiples, therefore, as .f is to Z?, so is Z^ to K. [v. i s] tti tA • BOOK V [v. 23 But, as ^ is to Z7, so is C to £' ; therefore also, as // is to A', so is C to B, [v. n] Again, since L, M are equimultiples of C, E, therefore, as C is to E, so is L to jIT. [v. 15] But, as C is to E, sovsHtoK; therefore also, as H is to K, so is L to M, (v. 1 1] and, alternately, as //^ is to Z, so is A' to M. [v. 16] But it was also proved that, ^s G \s to H, so is M to A''. Since, then, there are three magnitudes G, H, L, and others equal to them in multitude K, M, N, which taken two and two together are in the same ratio, and the proportion of them is perturbed, therefore, ex aequalif if G is in excess of L, K is also in excess of A''; , ., , ^., if equal, equal; and if less, less. [v. n] And G, K are equimultiples of A, D, and L, N oi C, F. Therefore, as A is to C, so is Z? to /^ Therefore etc. Q. E. D. There b an important difference between the version given by Simson of one part of the proof of this proposition and that found in the Greek text of Heiberg. Peyrard's ms. has the version given by Heiberg, but Simson's version has the authority of other mss. The Basel editw prinaps gives both versions (Simson 's being the first). After it has been proved by mean? of V. 1 5 and V. 1 1 that, as G is to jff, so is jV to A^ or, with the notation used in the note on Prop, *o, ma \ mb = ne ; nf, it has to be proved further that, • as .ff is to Z, so is /T to M, or mb •.nc = md : ne, and it is clear that the latter result may be directly inferred from v. 4, The reading translated by Simson makes this inference : " And because, as £ is to C, so is Z? to £, and H, K sxt equimultiples of S, /?, and L, Mot C, E, therefore, as H is to Z, so is K to M" [v. 4] The version in Hei berg's text is not only much longer (it adopts the roundabout method of using each of three Propositions v. 11, 15, 16 twice V. 33. S4] PROPOSITIONS 33, 24 183 over), but it is open to the objection that it uses v, 1 6 which is only applicable if the four magnitudes are of the same kind; whereas v. 33, the proposition now in question, is not subject to this restriction. Simson rightly observes that in the last step of the proof it should be stated that " G, K are any equimultiples whatever of A, D, a.nA L, N any whatever of C, F." He also gives the extension of the proposition to any number of magnitudes, enunciating it thus 1 " If there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio ; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last " ; -,.••,,'. and adding to the proof as follows : \ . t < 1 "Next, let there be four magnitudes A, B, C, D, and other four E, F, G, If, which, taken two and two in a cross order, have the same ratio, viz. : Am BasGto H, Bto Cas FioG, A B C -W6A CtoDasM toF; | E F Q H then A is to J} a& £ to M. Because A, B, C are three magnitudes, and F, G, H other three which, taken two and two in a cross order, have the same ratio, by the first case, .^ is to C as ^to H. Ii ^ <• But C is to .0 as £ is to F\ wherefore again, by the first case, . , A\s,\oDi&E\oH. And so on, whatever be the number of magnitudes." Proposition 24. If a first magnitude have to a second the same ratio as a third has to a fourth, and also a fifth have to the second the same ratio as a sixth to the fourth, the first and fifth added together milt have to the second the same ratio as the third and sixth have to the fourth. Let a first magnitude AB have to a second C the same ratio as a third DE has to a fourth F; f^ — : g q and let also a fifth BG have to o the second C the same ratio as d 1 H a sixth EJ/ has to the fourth f E; 1 say that the first and fifth added together, AG, will have to the second C the same ratio as the third and sixth, I? J/, has to the fourth E. 1*4 BOOK V [v. 34 For since, as BG is to C, so is B// to F, inversely, as C is to BG, so is ^ to BH. Since, then, as AB is to C, so is DB to B, and, as C is to BG, so is F to B//, therefore, ex aequali, as AB is to ^G, so is DB to ^/^. [v. n] And, since the magnitudes are proportional separando, they will also be proportional componendo ; [v. 18] therefore, as AG is to GB, so is Z?/^ to HB. But also, as BG is to C, so is BH to A ; therefore, ex aegtta/i, as ^4 6" is to C, so is DH to F. [v. aa] Therefore etc. Q. e. d. Algebraically, if a -.c = d:f, and b\(~e:f, then ■• {a-\-b):c={d-¥i):J. This profwsition is of the same character as those which precede the propositions relating to compounded ratios ■ but it could not be placed earlier than it ts because v. 22 is used in the proof of it. Inverting the second proportion to c\b-f:t, it follows, by v, 23, that a;6 = d:t, whence, by v. 18, {a + b):i ~{d-i-e) : e, and from this and the second of the two given proportions we obtain, by a fresh application of v. 22, {a-^l,):c=(d*e):/. The first use of v. as is important as showing that the opposite process to compounding ratios, or what we should now call division of one ratio by another, does not require any new and separate propositions. Aristotle tacitly uses v. 24 in combination with v. 1 1 and v, 16, Meleorologica '"- S> 37^a 22 — 26. Simson adds two corollaries, one of which (Cor. 3) notes the extension to any number of magnitudes. " The proposition holds true of two ranks of magnitudes whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to a fourth magnitude ; as is manifest" Simson's Cor. i states the corresponding proposition to the above with separando taking the place of compomnds, viz., that corresponding to the algebraical form {a-b);c^{fi-e):f. "Cor. I. If the same hypothesis be made as in the proposition, the excess of the flrst and fifth shall be to the second as the excess of the third and sixth to the fourth. The demonstration of this is the same with that of the proposition if division be used instead of composition." That is, we use V. 17 instead of v. 18, and conclude that {a~b):b = {d-t):t. V. as] PROPOSITIONS »4, »s ^ Proposition 25. If four magnitudes be proportional, the greatest and the least are greater than the remainittg two. Let the four magnitudes A£, CD, E, F be proportional so that, as AB is to CD, so is E to F, and let AB be the greatest of them and F the least ; ^ Q b I say that AB, F are greater than c CD, E. H P c 1 — For let AG be made equal to E, , and CH equal to F, Since, as ^^ is to CD, so is ^ to F, and E is equal to AG, and /^ to CM, therefore, as AB is to CD, soh AG to C/T. And since, as the whole AB is to the whole CD, so is the part AG subtracted to the part C// subtracted, the remainder GB will also be to the remainder HD as the whole AB is to the whole CD. [v. 19] But AB is greater than CD ; therefore GB is also greater than HD. And, since AG\& equal to E, and CH to F, therefore AG, F a.re equal to CH, E. And if, GB, HD being unequal, and GB greater, AG, F be added to GB and C^, ^ be added to HD, it follows that AB, F are greater than CD, E. Therefore etc. Q. E. D. Algebraically, if a-.b^cid, and a is the greatest of the four magnitudes and d the least, a-¥ d> 6 + c. Simaon is right in inserting a word in the setting-out, "let AB be the greatest of Ihem and <censequenily> J' the least." This follows from the particular case, really included in Def. 5, which Sinison makes the subject of his proposition A, the case namely where the equimultiples taken are ante the several magnitudes. The proof is as follows. Since a:b = £:d, a — {\h — d=a:b, [v. 19] iM BOOK V [v. 25 But «>*; therefore (tf-^)>(*-i/). • [v. 16 and 14] Add to each {c+d); therefore (o + rf) > (i + c). There is an important particular case of this proposition, which is, however, not mentioned here, vh. the case where * = c. The result shows, in this case, that tAe arithmttic PKan between two magnitudes is greater than ffielr geometric mean. The truth of this is proved for straight lines in vi.- 27 by "geometrical algebra," and the theorem forms the Siopur/ioj for equations of the second degree. Simson adds at the end of Book y, four propositions, F, G, H, K, which, however, do not seem to be of sufficient practical use to justify their inclusion here. But he adds at the end of his notes to the Book the following paragraph which deserves quotation word for word. "The 5th book being thus corrected, I most le.idily agree to what the learned Dr Barrow says, 'that there is nothing in ihe whole boily of the elements of a more subtile invention, nothing more solidly established, and more accurately handled than the doctrine of proportionals.' And there is some ground to hope that geometers will think that this could not have been said with as good reason, since Theon's time till the present." Simson's claim herein will readily be admitted by all readers who are competent to form a judgment upon his criticisms and elucidations of Book V. BOOK VI. INTRODUCTORY NOTE. The theory of proportions has been established in Book v. in a perfectly general form apphcable to all kinds of magnitudes (although the representation of magnitudes by straight lines gives it a j^eo metrical appearance) ; it is now necessary to apply the theory to the particular case oi geometrical investigation. The only thing still required in order that this may be done is a proof of the existence of such a magnitude as bears to any given finite magnitude any given finite ratio ; and this proof is supplied, so far as regards the subject matter of geometry, by vi. n which shows how to construct a fourth pro- portional to three given straight lines, A few remarks on the enormous usefulness of the theory of proportions to geometry will not be out of place. We have already in Books i. and ii. made acquaintance with one important part of what has been well called geometrical algebra, the method, namely, of application of areas. We have seen that this method, working by the representation of products of two quantities as rectangles, enables us to solve some particular quadratic equations. But the limitations of such a method are obvious. So long as general quantities are represented by straight lines only, we cannot, if our geometry is plane, deal with products of more than two such quantities ; and, even by the use of three dimensions, we cannot work with products of more than three quantities, since no geometrical meaning could be attached to such a product. This limitation disappears so soon as we can represent any general quantity, corresponding to what we denote by a letter in algebra, by a ratio; and this we can do because, on the general theory of proportion established in Book v., a ratio may be a ratio of two incommensurable quantities as well as of com mensu rabies. Ratios can be compoundeti ad infinitum, and the division of one ratio by another is equally easy, since it is the same thing as compounding the first ratio with the inverse of the second. Thus e,g. it is seen at once that the coefficients in a quadratic of the most general form can be represented by ratios between straight lines, and the solution by means of Books i. and n, of problems corresponding to quadratic equations with particular coefficients can now be extended to cover any quadratic with real roots. As indicated, we can perform, by composition of ratios, the operation corresponding to multiplying algebraical quantities, and this to any extent. We can divide quantities by compounding a ratio with the inverse of the ratio representing the divisor. For the addition and subtraction of quantities we have only to use the geometrical equivalent of bringing to a common denominator, which is effected by means of the fourth proportional. .. ^ . , _ i8» BOOK VI [vi. DErr. DEFINITIONS. I. Similar rectilineal figures are such as have their angles severally equal and the sides about the equal angles proportional. [2. Reciprocally related figures. See noie.'\ ■'■•'' 3. A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less. 4. The height of any figure is the perpendicular drawn from the vertex to the base. Definition i. QfiGija. (Tj^^^ara tv&vypofifid fiTTtv, wra rav re ymvia^ l^tK i\Vr Kara fjttav ital T&f TTtpl Tat tirat '^ott'taf frX^pa^ (ikoAd^OK T'lis definition is quoted by Aristotle, Ana/, post. 11. 17, 99 a 13, where he says that simUantjf (to o;u.Diof) in the case of figures "consists, let us say (htuk), in their having their sides proportional and their angles equal." The use of the word laat may suggest that, in Aristotle's time, this definition had not quite established itself in the text-books (Heibeig, MathemaiUches zu Arisiolties, p. g). It was pointed out in Van Swinden's Eknunts of Geometry (Jacobi's edition, 1834, pp. 1 14 — 5) that Ei^uclid omits to stale an essential part of the definition, namely that "the corresponding sides must be opposite to equal angles," which is necessary in order that the corresponding sides may follow in the same order in both figures. At the same time the definition states more than is absolutely necessary, for it is true to say that iwo polygons are similar when, if the iides and angles are taken in the same order, the angles are equal and the sides about the equal angles are proportional, omitting (i) three consecutive angles, or (2) two consecutive angles and the side common to them, or (3) two consecutive sides and the angle included by them, and making no assumption with regard to the omitted sides and angles. Austin objected to this definition on the ground that it is not obvious that the properties (i) of having their angles respectively equal and (2) of having the sides about the equal angles proportional can coexist in two figures ; but, a definition not being concerned to prove the existence of the thing defined, the objection falis to the ground. We are property left to satisfy ourselves as to the existence of similar figures in the course of the exposition in Book vi., where we learn how to construct on any given straight line a rectilineal figure similar to a given one (vi. i8j. VI. DEFF. a— s] DEFINITIONS 189 Definition 2. The Greek text gives here a. definition of riciprocally related fibres (dtTTnrtTTOV^d™ ay^^T<i\. "[Two] figures are redprocally relaUd when there are in each of the two figures antecedent and consequent ratios" ('An-HrnrokSora Si o-j()j/iaTa i<mv, oray tv iKorifxf Tiui' (j}(T)iiaTiati ijyiivfio'oi t€ nai Iro/uTOt Aoyoi wiTtv). No intelligible meaning can be attached to "antecedent and con- sequent ratios " here ; the sense would require rather " an antecedent and a consequent of (two equal) ratios in each figure." Hence Candalla and Peyrard read Xoywc Spot ("terms of ratios") instead of Ao'yiii. Camerer reads Xayar without upoi. But the objection to the definition lies deeper. It is never used; when we come, in vi. 14, 15, xi. 34 etc. to jjaiallelograms, triangles etc. having the property indicated, they are not called " reciprocal " parallelograms etc., but parallelograms etc. "/A^ sides ofwhUh are reciprocally proportional," w/ ot-r(irtiroi'Sa<ric at irAtupoi, Hence Sim son appears to be right in condemning the definition; it may have been interpolated from Heron, who has it. Simson proposes in his note to substitute the following definition. "Two magnitudes are said to be reciprocally proportional to two others when one of the first is to one of the other magnitudes as the remaining one of the last two is to the remaining one of the first." This definition requires that the magnitudes shall be all of the same kind. Definition 3. 'Afipav nat, fiiirov Aoyoi' tiStia Tir/iigcrAu Xiytmt, Srav ^ wt 7 Skii -rp^ rd fut^otf T/jiTJ/juij oi/ru? T^ fArti^ov irpof ra IXaTTCv* Definition 4. 'Ayofimj^ The definition of " height " is not found in Campanus and is perhaps rightly suspected, since it does not apply in terms to parallelograms, parallele- pipeds, cylinders and prisms, though it is used in the Elements with reference to these latter figures. Aristotle does not appear to know altitude (v^ot) in the mathematical sense; he uses naStTiK of triangles (Meiearelo^ea tn. 3, 373 a 11). The term is however readily understood, and scarcely requires definition. [Definition j. Aoyoc Ik koywv (TvyKturBai XcycTat, crap at twv koytav injAuroTi/rc; j0* lavrac woXXankatTiauSturat vtumtrl riva. "A ratio is said to be compounded of ratios when the sizes (jnjAiKortjrt?) of the ratios multiplied together make some (? ratio, or size),"] As already remarked (pp. 116, 132), it is beyond doubt that this definition of ratio is interpolated. It has little MS. authority. The best MS. (P) only has it in the margin; it is omitted altogether in Campanus' translation from the ijli* BOOK VI [vi. OEF. s Arabic ; and the other mss. which contain it do not agree in the position which they give to it. There is no reference to the definition in the place where compound ratio is mentioned for the first time (vi. t^), nor anywhere else in Euclid; neither is it ever referred to by the other great geometers, Archimedes, Apollonius and the rest. It appears to be only twice mentioned at all, {:) in the passage of Eutocius referred to above (p. ii6) and (») by Theon in his commentary on Ptolemy's ffun-o^u. Moreover the content of the definition is in itself suspicious. It speaks of the " sizes of ratios being multiplied together (literally, into themselves)," an operation unknown to geometry. There is no wonder that Eutocius, and apparently Theon also, in their efforts to explain it, had to give the word jrrfAiiHjnjt a meaning which has no application except in the case of such ratios as can be expressed by numbers (Eutocius e.g. making it the "number by which the ratio is called"). Nor is it surprising that Wallis should have found it necessary to substitute for the " quantitas " of Commandinus a different translation, " quantuplicity," which he said was represented by the "expeneni af the ratio" ( ratio nis ex- ponens), what Peletarius had described as "denominatio ipsae pro portion! s" and Clavius as "denominator." The fact is that the definition is ungeometrical and useless, as was already seen by Savile, in whose view it was one of the two blemishes in the body of geometry (the other being of course Postulate 5). It is right to add that Hultsch (art. "Eukleides" in Pauly-Wissowa's Real- EtuytlopddU dtr danischen Allertumswissenschaft) thought the definition genuine. His grounds are (i) that it stood in the iroAaui ln&ooi^ repre- sented by P (though P has it in the margin only) and (a) that some ex- planation on the subject must have been given by way of preparation for VI. 25, while there is nothing in the definition which is incomisteni with the mode of statement of vi. 23. If the definition is after all genuine, I should be inclined to regard it as a mere survival from earlier textbooks, like the first of the two alternative definitions of a solid angle (xt, Def 11); for its form seems to suit the old theory of proportion, applicable to commensurable magnitudes only, better than the generalised theory of Eudoxus, v. kI|.'-' . ..i \-!\' .... ;*-,, r,. jV 1,'r.i' -■ ' '" BOOK VI. PROPOSITIONS. Propositiok '•> J MiS' Triangles and parallelograms which are under the same height are to one another as their bases. Let ABC, A CD be triangles and EC, C/^ parallelograms under the same height ; j I say that, as the base BC is to the base CZ7, so is the triangle ABC to the triangle A CD, and the parallelogram £C to the parallelogram CF. For let BD be produced in both directions to the points ff, L and let [any number of straight lines] BG, GH be lo made equal to the base BC, and any number of straight lines DK, KL equal to the base CD ; let AG, AH, AK, AL be joined. Then, since CB, BG, GH are equal to one another, the triangles ABC, AGB, AHG are also equal to one IS another. [i. 38] Therefore, whatever multiple the base HC is of the base BC, that multiple also is the triangle AHC of the triangle ABC. For the same reason, 10 whatever multiple the base ZC is of the base CD, that multiple also is the triangle ALC of the triangle ACD ; and, if the base HC is equal to the base CL, the triangle AHC is also equal to the triangle ACL, [i. 38] r9» BOOK VI [n. i if the base //C is in excess of the base CZ., the triangle AHC as is also in excess of the triangle A CL, and, if less, less. Thus, there being four magnitudes, two bases BC, CD and two triangles ABC, ACD, equimultiples have been taken of the base BC and the 30 triangle ABC, namely the base HC and the triangle AHC, and of the base CD and the triangle ^Z?C other, chance, equi- multiples, namely the base LC and the triangle ALC \ and it has been proved that, if the base HC is in excess of the base CL, the triangle AHC 3S is also in excess of the triangle ALC ; if equal, equal ; and, if less, less. Therefore, as the base BC is to the base CD, so is the triangle ABC to the triangle ACD. [v. Def. 5] Next, since the parallelogram EC is double of the triangle AoABC, [i. 4>] and the parallelogram FC is double of the triangle ACD, while parts have the same ratio as the same multiples of them, [v. 15] therefore, as the triangle ABC is to the triangle ACD, so is 45 the parallelogram £C to the parallelogram FC. Since, then, it was proved that, as the base BC is to CD, so is the triangle ABC to the triangle A CD, and, as the triangle ABC is to the triangle ACD, so is the parallelogram EC to the parallelogram CF, 50 therefore also, as the base BC is to the base CD, so is the parallelogram EC to the parallelogram FC. [v. n] Therefore etc. Q, E. D. 4. Under the same height. The Greek text has "under Ihe iame height AC," with a figure in which the side ^C commun to the two triangles is perpendicular to the base and is therefore iudf the "height." But, even if tlie two triangles are placed contiguously so as to have a commbn side AC, it is quite gratuiicnis to require it to be perpendicular to the base. Theon, on this occasion making an improvement, altered to " which are [Ato) under the same height, (namely! 'he perpendicular drawn from A to BD," I iiave vetitured lo alter so far as lo omit "AC" and to draw the figure in the usual way. 14. ABC.AGBiAHG. Euclid,indiaferenttoeitactorder,writes" AffG, AGB,ABC." 46. Since then it was proved that, as the base BC i( to CD, *o It the triangle ABC lo the triangle ACD. Here again words have to be supplied in translatin|> the eitremely terse Greek irtl a^ ^Htlx^fj "• 1'^' ^ fii'" Br rpit T^e Ti, otrut ri ABr Tf/iyuror r^i ri ATA Tplywrwr, literSly " since was proned, as the base BC to CO, to the truaigle ABC lo Ihe triangle ACJi." Cf. note on v. 16, p. i6j. VI. i] PROPOSITION I 193 The proof assumes — what is however an obvious deduction from 1. 38 — that, of triangles or parallelograms on unequal bases and between the same parallels, the greater is that which has the greater base. It is of course not necessary that the two given triangles should have a common side, as in the figure ; the proof is just as easy if they have not. The proposition being equally trtie of triangles and parallelograms of eqital heights, Simson states this fact in a corollary thus: " From this it is plain that triangles and parallelograms that have equal altitudes are to one another as their bases. Let the figures be so placed as to have their bases in the same straight line ; and, if we draw perpendiculars from the vertices of the triangles to the basw, the straight line which joins the vertices is parallel to that in which their bases are, because the perpendiculars are both equal and parallel to one another [i, 33]. Then, if the same construction be made as in the proposition, the demonstration will be the same." The object of placing the bases in one straight line is to get the triangles and parallelograms within (hi same parallels. Cf. Proclus' remark on i. 38 (p. 405, 17) that having the same height is the same thing as being in the same parallels. Rectangles, or right-angled triangles, which have one of the sides about the right angle of the same length can be placed so that the equal sides coincide and the others are in a straight line. If then we call the common side the base, the rectangles or the right-angled triangles are to one another as their heights, by vi. i. Now, instead of each right-angled triangle or rectangle, we can take any other triangle or parallelogram respectively with an equal base and between the same parallels. Thus Triangles and paralklograms having eguai bases art to one another as their heights. Legendie and those authors of tnodem text-books who follow him in basing their treatment of proportion on the algebraical definition are obliged to divide their proofs of propositions like this into two parts, the first of which proves the particular theorem in the case where the magnitudes are commensurable, and the second extends it to the case where they are incommensurable. L^endre (Aliments dt G'eometrie, \\\. 3) uses for this extension a rigorous method by reductie ad absurdum similar to that used by Archimedes in his treatise On the equilibrium 0/ planes^ 1. 7. The following is Legendre's proof of the extension of vi, i to in- commensurable parallelograms and bases. The proposition having been proved for commensurable bases, let there be two rectangles AS CD, AEFD as in the figure, on bases AB, .^£which are incommensurable with one another. To prove that recL A BCD: recL AEFD =AB: AE. For, if not, let red. A3CD -rect. AEFD = AB : AO, (ij where AO h (for instance) greater than AE. Divide AS into equal parts each of which is less than EO, and mark off on AO lengths equal to one of the parts; then there will be at least one point of division between E and O. -r Let it be /, and draw /AT parallel to EF. * «94 BOOK VI [vi. 1, » Then ihe rectangles A BCD, AIKD are in the ratio of the bases AB, A I, since the latter are commensurable. Therefore, inverting the proportion, rect. AIKD:^<ixx. ABCD'^AI.AB (i). From this and (i), « atquati, rect, AIKD : rect. AEFD = AI.AO. , _ But A0> A/; therefore rect. AEBD>Ttci. AIKD. But this is impossible, for the rectangle AEFD is less than the rectangle AIKD. Similarly an impossibility can be proved \i AO < AE. > Therefore lecL ABCD : rect. AEFD = AB : AE. Some modern American and German text-books adopt the less rigorous method of appealing to the theory of iimiis. Proposition 2. If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally ; and, if Ike sides of the triangle be cut proportionally, the line joining the points of section will be parallel to the remaining side of the triangle. For let DE be drawn parallel to BC, one of the sides of the triangle ABC; I say that, as BD is to DA, so is CE to EA. For let BE, CD be joined. Therefore the triangle BD E is equal to the triangle CDE ; for they are on the same base DE and in the same parallels DE, BC. [1. 38] And the triangle ADE is another area. But equals have the same ratio to the same ; [v. 7] therefore, as the triangle BDE is to the triangle ADE^ so is the triangle CDE to the triangle ADE, But, as the triangle BDE is to ADE, so is BD to DA ; for, being under the same height, the perpendicular drawn from E to AB, they are to one another as their bases, [vi. i] For the same reason also, as the triangle CDE is to ADE, so is CE to EA. '- Therefore also, as BD is to DA, so is CE to EA. [v. n] VI. a, 3] PROPOSITIONS 1—3 195 Again, let the Sides AB, AC oi the triangle ABC be cut proportionally, so that, as BD is to DA, so is C£ to EA ; and let DE be joined. I say that DB is parallel to BC - f •" • For, with the same construction, i ■ •"> -^ • since, as BD is to DA, so is C£ to £A, but, as BD is to DA, so is the triangle BDE to the triangle ADE, and, as CE is to EA, so is the triangle CDE to the triangle ADE, [v.. ,] therefore also, as the triangle BDE is to the triangle ADE, so is the triangle CDE to the triangle ADE. [v. n] Therefore each of the triangles BDE, CDE has the same ratio to ADE. Therefore the triangle BDE is equal to the triangle CDE\ [V.9] and they are on the same base DE. But equal triangles which are on the same base are also in the same parallels. [i. 39] Therefore DE is parallel to BC. Therefore etc. Q, E. D. Euclid evidently did not think it worth while to distinguish in the enunciation, or in the figure, the cases in which the parallel to the base cuts the othei two sides produced (a) beyond the point in which they intersect, (i) m the other direction. Simson gives the three figures and inserts words in the enunciation, reading "it shall cut the other sides, or those lidts produced, proportionally" and "if the sides, or the sidts produced, be cut proportionally." Todhunter observes that the second part of the enunciation ought to make it clear which segments in the proportion correspond to which. Thus e.g., if AD were double of DB, and CE double of EA, the sides would be cut proportionally, but DE would not be parallel to BC. The omission could be supplied by saying "and if the sides of the triangle be cut proportionally io that tht segments adjacent to the third side «ri corresponding terms in the proportion." .....-.,■ i ' |« , . ..... .,. Proposition 3. . // an angle of a triangle be bisected and the straight line cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle; and, if ike segments of the base have the same ratio as the 1^6 - BOOK VI [vi. 3 remaining sides of the triangle, the straight line joined from the vertex to the point of section will bisect the angle of the triangle. Let ABC be a triar^le, and let the angle BA C be bisected by the straight line AD \ I say that, as BD is to CD, so is BA to AC. For letC^ be drawn through C parallel to DA, and let BA be carried through and meet it Then, since the straight line A C falls upon the parallels AD, EC, the angle ACE is equal to the angle CAD. [i, 39] But the angle CAD is by hypothesis equal to the angle BAD; therefore the angle BAD is also equal to the angle ACE. Again, since the straight line BAE falls upon the parallels AD, EC, the exterior angle BAD is equal to the interior angle A EC. [i. 39] But the angle ACE was also proved equal to the angle BAD; therefore the angle A CE is also equal to the angle A EC, so that the side AE is also equal to the side AC. [i. 6] And, since AD has been drawn parallel to EC, one of the sideij of the triangle BCE, therefore, proportionally, as BD is to DC, so is BA to AE. But AE is equal to AC; t^'- '] therefore, as BD is to DC, so is BA to AC. Again, let BA be to ^Cas BD to DC, and let AD be joined ; I say that the angle BAC has been bisected by the straight line AD. For, with the same construction, since, as BD is to DC, so is BA to AC, •'■ -^^ i' • ^<y-^ n.3] PROPOSITION 3 197 and also, as BD is to DC, so is BA to AE\ for AD has been drawn parallel to EC, one of the sides of the triangle BCE : [VI. 2] therefore also, as BA is to A C, so is BA to AE. [v. i r] Therefore AC is equal to AE, [v. 9] so that the angle A EC is also equal to the angle ACE. [i. 5] But the angle A EC is equal to the exterior angle BAD, [I. jg] and the angle ACE is equal to the alternate angle CAD; ['d.] therefore the angle BAD is also equal to the angle CAD. Therefore the angle ^^C has been bisected by the straight line AD. Therefore etc. •' • Q, E. D. The demonstration assumes that C£ will meet BA produced in some point £. This is proved in the same way as it is proved in vi. 4 that BA, ED will meet if produced. The angles ABD, SDA in the figure of vc, 3 are together less than two right angles, and the angle BDA is equal to the angle BCE, since DA, CE are parallel. Therefore the angles ABC, BCE are together less than two right angles ; and BA, CE must meet, by 1. Post. 5. The corresponding proposition about the segments into which ^C is divided externally by the bisector of the external angU at A when that bisector meets BC produced (i.e. when the sides AB, AC ak not equal) is important. Simson gives it as a separate proposition, A, noting the fact that Pappus assumes the result without proof (Pappus, vii. p, 730, 24). The best plan ts however, as De Morgan says, to combine Props. 3 and A in one proposition, which may be enunciated thus : If an angle of a triangle be bisected internally or externally by a straight line which cuts the opposite side or the opposite side produced, the segments of that side will have the same ratio as the other sides of the triangle; and, if a side of a triangle be divided internally or externally so that its segments have the same ratio as the other sides of the triangle, the straight line drawn from the point of section to the angular point which is opposite to the first ?nentiomd side will bisect the interior or exterior angle at that angular point. Let ..4 C be the smaller of the two sides AB, AC, so that the bisector AD of the exterior angle at A may meet BC produced beyond C. Draw CE through C parallel to DA, meeting BA in E. Then, if EA C is the exterior angle bisected by AD in the case of external bisection, and if a point Eis taken on AB io the figure of vi. 3, the proof of t9». BOOK VI [vi. 3 VL 3 can be used almost word for word for the other ease. We have only to spak of the angle "/^^^C" for the angle " BAC," and of the angle "FAD" for the angle " BAD " wherever they occur, to say "let SA^ or BA produced, meet CE in E," and to substitute " BA or BA produced" for "BAE" lower down. . h If AD, AE be the internal and external bisectors of the angle A in a. triangle of which the sides AB, AC are unequal, AC being the smaller, and if AD, AE meet BC and BC produced \n D, E respectively, the ratios of BD to i>Cand of BE to EC are alike equal to the ratio of BA to AC. Therefore BE is to ECtis BD to DC, that is, BE is to EC as the difference between BE and ED is to the difference between ED and EC, whence BE, ED, EC are in karmonit prograsion, or DE is a harmonic mtan between BE and EC, or again B, D, C, £ is a harm^nu range. Since the angle DAC is half of the angle BAC, and the angle CAE half of the angle CAF, while the angles BAC, CAF are equal to two right angles, the angle DAE is a right angle. Hence the circle described on DE as diameter passes through A. Now, if the ratio of BA to ^Cis given, and if BC is given, the points D, E on BC and BC produced are given, and therefore so is the circle on D, E as diameter. Hence M« /acus of a point sueh that its dittanas from two given points are in a given ratio {net being a ratio of equality) is a rirck. This locus was discussed by ApoUonius in his Plane Loci, Book ii., as we know fr«m Pappus (vii, p. 666), who says that the book contained the theorem that, if from two given jwjnts straight lines inflected to another point are in a given ratio, the point in which they meet will lie on either a straight line or a circumference of a circle. The straight line is of course the locus when the ratio is one of equality. The other case is quoted in the following form by Eutocius (ApoUonius, ed. Heiberg, ti. pp. \ 80—4). Given two points in a plane and a proportion between unequal strwght lines, it is possible to describe a circle in the plane so that the straight lines inflected from the given points to the dreumference of the circle shall have a ratio the tame as the given one. ApoUonius' construction, as given by Eutocius, is remarkable because he makes no use of either of the points D, E. He finds 0, the centre of the required circle, and the length of its radius directly from the data BC and the given ratio which we will call h : k. But the construction was not discovered by ApoUonius j it belongs to a much earlier date, since it appears in exactly VI. j] PROPOSITION 3 199 the same rorm in Aristotle, Mekorolegica in. 5, J76 a 3 sqq. The analysis leading up to the construction is, as usual, not given either by Aristotle or Eutocius. We are told to take three straight lines x, CO (a length measured along BC produced beyond C, where J, C are the points at which the greater and smaller of the inflected lines respectively terminate), and r, such that, if h\khi the given ratio and h>k, k:h = hik + x, (a) ■ '' x:SC=k:CO = h:r 08) This determines the position of O, and the length of r, the radius of the required circle. The circle is then drawn, any point P is taken on it and joined to B, C respectively, and it is proved that FB'.PC^h.k. We may conjecture that the analysis proceeded somewhat as follows. Ft would be seen that £, C are "conjugate points" with reference to the circle on DE as diameter. (Cf. ApoUonius, Cmks, i. 36, where it is proved, in terms, for a circle as well as for an ellipse and a hyperbola, that, if the polar of j5 meets the diameter DE in C, then EC: CD = EB : BD.) If O be the middle point of DE, and therefore the centre of the circle, D, E may be eliminated, as in the Conies, i. 37, thus. Since EC : CD = EB : BD, it follows that EC+CD: EC~ CD = EB ■¥ BD -.EB- BD, or iOD : 20C= tOB : tOD, '«■ ' that is, BO.OC= OD' = r', say. If therefore B be any point or the circle with centre O and radius r, BO: OP=OP:OC, so that 50/", PO Care similar triangles. ., .,, , ... \a.3A6:*:\on,h:k-BD:DC = BE-^EC ' ' •• 11 • • ^BD + B£:D£ = BO:r. •• •'• ■-'"' Hence we require that BO:r = r:OC=BP:PC=h.k (8) Therefore, alternately, k:CO^h:r, which is the second relation in (^) above. '• ' Now assume a length x such that each of the last ratios is equal \ax\BC, as in (^). MiO BOOK VI [vi. 3, 4 Then ' • . x:BC-k:CO = h:r. Therefore .r + A : BO -h:r, and, alternately, x + k:h = BO : r I =hik, from (S) above ; and this is the relation (a) which remained to be found. ApoUonius' proof of the construction is given by Eutocius, who begins by saying that it is manifest that r is a mean proportional between BO and OC. This IS seen as follows . From (j8) we derive x\BC=k: CO = A:r = {k-yx):BO, whence BO ■.r={k + x):h m:A:A, by <a), = r:CO, by(^), and therefore r* = BO . CO. But the triangles BOP, POC have the angle at common, and, since BO: OP = OF: OC, the triangles are similar and the angles OPC, OBP are equal [Up to this point Aristotle's proof is exactly the same ; from this point it diverges slightly.] If now CL be drawn parallel to BP meeting OP in L, the angles BPC ZCP are equal also. Therefore the triangles BPC, PCI. are similar, and BP:PC=PC:CL, whence BP^.PC^^BP: CL <= BO : OC, by parallels, = BO^ : OP* (sincere ; OP= OP: OC). Therefore BF:PC^BO:OP = A;*(for OP=-r). [Aristotle infers this more directly from the similar triangles P03, COP. Since these triangles are similar, OP: CP=OB:BP, whence BP: PC'' BO :0P "• = h:k.'\ ApoUonius proves lastly, by reductio ad ahsurdum, that the last equation cannot be true with reference to any point P which is not on the circle so described. Proposition 4. In equiangular triangles the sides about (he equal angles are proportional, and those are corresponding sides which subtend the equal angles. VI. 4] PROPOSITIONS 3, 4 sol Let ABC, DCE be equiangular triangles having the angle ABC equal to the angle DCE, the angle BAC to the angle CDE, and further the angle ACB to the angle CED ; I say that in the triangles ABC, DCE the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles. For let BC be placed in a straight line with CE. Then, since the angles ABC, ACB are less than two right angles, [i, 17] and the angle ACB is equal to the angle DEC, therefore the angles ABC, DEC are less than two right angles ; therefore BA, ED, when produced, will meet. [i. Post. 5] Let them be produced and meet at E, Now, since the angle DCE is equal to the angle ABC, BF is parallel to CD. [1. *8] Again, since the angle ACB is equal to the angle DEC, AC is parallel to FE. [i. 38] Therefore FACD is a parallelogram ; therefore FA is equal to DC, and AC to FD. fi. 34] And, since AC has been drawn parallel to FE, one side of the triangle FBE, therefore, as BA is to AF, so is BC to CE. [vi. 1] But AF is equal to CD ; therefore, as BA is to CD, so is BC to CE, and alternately, as AB is to BC, so is DC to CE. [v. 16] '1 Again, since CD is parallel to BF, ' therefore, as BC is to CE, so is FD to DE. [vi. »] But FD is equal to AC; therefore, as BC is to CE ^o\^ AC to DE, and alternately, as BC is to CA, so is CE to ED. [v. r6] ■M3 BOOK VI [VI. 4, s Since then it was proved that, ' as AB is to BC, so is DC to CE, and, as BC is to CA, so is CE to ED ; therefore, ex aeguali, as BA is to ^C, so is CD to DE. [v. sa] Therefore etc. Q, E. D, Todhunter remarks that " the manner in which the two triangles are to be placed is very imperfectly described; their bases are to be in the same straight line and contiguous, their vertices are to be on the same side of the base, and each of the two angles which have a common vertex is to be equal to the remote angle of the other triangle," But surely Euclid's description is sufficient, except for not saying that B and D must be on the same side of BCE. VI. 4 can be immediately deduced from vi. a if we superpose one triangle on the other three times in succession, so that each angle successively coincides with its equal, the triangles being similarly situated, e.g. if {A, B, C and D, E, F being the equal angles respectively) we apply the angle DEFio the angle ABC so that D lies on AB {produced if necessary) and J^on BC (produced if necessary). De Morgan prefers this method. " Abandon," he says, " the peculiar mode of construction by which Euclid proves two cases at once; make an angle coincide with its equal, and suppose this process repeated three times, one for each angle." . , Proposition 5. If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend. Let ABC DEF be two triangles having their sides proportional, so that, as AB is to BC, so is DE to EF, sis BC is to C A, so is EF to FD, • ' ' and further, as BA is to AC, so is ED to DF; I say that the triangle ABC is equiangular with the triangle DEE, and they will have those angles equal which the corre- sponding sides subtend, namely the angle ABC to the angle DEE, the angle BCA to the angle EFD, and further the angle BAC to the angle EDF. For on the straight line EF, and at the points E, F on it, let there be constructed the angle FEG equal to the angle ABC, and the angle EFG equal to the angle A CB ; [i. aj] n. s] PROPOSITIONS 4. S Bp therefore the remaining angle at A is equal to the remaining angle at G. [t- 3*] Therefore the triangle ABC is equiangular with the triangle G£F. Therefore in the triangles ABC, GEF the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles ; [vi. 4] therefore, as AB is to BC, so is GE to EF. But, as AB is to BC, so by hypothesis is DE to EF\ therefore, as DE is to EF, so is GE to EF. [v. n] Therefore each of the straight lines DE, GE has the same ratio to EF; therefore DE is equal to GE. [v. g] For the same reason DF is also equal to GF. Since then DE is equal to EG, and EF is common, the two sides DE, EF are equal to the two sides GE, EF; and the base DF is equal to the base EG ; therefore the angle DEF is equal to the angle GEF, [1. 8] and the triangle DEF is equal to the triangle GEF, and the remaining angles are equal to the remaining angles, namely those which the equal sides subtend. [1. 4] Therefore the angle DEE is also equal to the angle GFE, and the angle EDF to the angle EGF. And, since the angle FED is equal to the angle GEF, while the angle GEF is equal to the angle ABC, therefore the angle ABC is also equal 10 the angle DEF. :.aB4 BOOK VJ • [vi. s, 6 For the same reason the angle ACB is also equal to the angle DFE^ "■■ and further, the angle at A to the angle at D ; therefore the triangle ABC is equiangular with the triangle DBF. Therefore etc. Q. E, D. This proposition is the complete converse, vi. 6 a partial converse, of vi. 4. Todhuntcr, after Walker, remarks that the enunciation should make it clear that the sides of the triangles laken in order are proportional. It is quite f»o&sible that there should be two triangles ABC, Z'£/'such that AB is to .SCas DE to EF, and .SCisto C^^as DF'xs, to EJ> (instead of ^i^to/Z)), so that A3 is to AC as J?Fto EF {fx aequali \n ptrlurbed prepartion)\ in this case the sides of the triangles are proportional, but not in the same order, and the triangles are not necessarily equiangular to one another. For a numerical illustration we may suppose the sides of one triangle to be 3, 4 and 5 feet respectively, and those of another to be 11, 15 and 10 feet respectively. In VI. 5 there is the same apparent avoidance of indirect demonstration which has been noticed on t. 48. i •■. .m u 1 ■ t ' Proposition 6. If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles will be equiangular and will have those angles equal which the corre- sponding sides subtend. Let ABC, DEF be two triangles having one angle BAC equal to one angle EDF and the sides about the equal angles proportional, so that, as BA is to AC, so is ED to DF\ I say that the triangle ABC is equiangular with the triangle DEF, and will have the angle ABC equal to the angle DEF, and the angle ACB to the angle DFE. For on the straight line DF, and at the points D, Fon it, let there be constructed the angle FDG equal to either of the angles BAC, EDF, and the angle DFG equal to the angle ACB; [1.23] therefore the remaining angle at B is equal to the remaining angle at G, -. ... . , » .. [•• 3'] VI. 6J PROPOSITIONS s, 6 ^6$ Therefore the triangle ABC is equiangular with the triangle DGF. Therefore, proportionally, as BA is to AC, so is GD to DF. [vi. 4] But, by hypothesis, a.sBA is to AC, so also is £/? to I}F; therefore also, as FD is to /)F, so Is GD to DF. [v. n] Therefore £D is equal to BG ; • • ' [-^ ^j and Z?/^ is common ; 1 • therefore the two sides £D, DFare equal to the two sides GD, DF; and the angle EDF is equal to the angle GDF; therefore the base EF is equal to the base GF, ' and the triangle DEF is equal to the triangle DGF, and the remaining angles will be equal to the remaining angles, namely those which the equal sides subtend. [i. 4] Therefore the angle DFG is equal to the angle DFE, and the angle DGF to the angle DEF. But the angle DFG is equal to the angle ACB; ; 1 li-- therefore the angle ACB is also equal to the angle DFE. And, by hypothesis, the angle BAC is also equal to the angle EDF; therefore the remaining angle at B is also equal to the remaining angle at E\ ^ ■ y .1 ['-3*] therefore the triangle ABC is equiangular with the triangle DEF. Therefore etc. • < cj. ■ . ■' ' ■ -' . ^ 1 ■ *• • ~ ' ;■> ■ Q. E, D. 006 <•> ,t BOOK VI [Ti, 7 Proposition 7. '•' •'•' ^'.f' ' 1/ two triangles have one angle equal to one angle, the sides about other angles proportional, and th-e remaining angles either both less or both not less than a right angle, the triangles will be equiangular and will have those angles equal, the stdes about which are proportional. Let ABC, DEFhe two triangles having one angle equal to one angle, the angle BAC to the angle EJDF, the sides about other angles ABC, DEF propor- tional, so that, as AB ts to BC, so is DE to EF, and, first, each of the remaining angles at C, F less than a right angle ; I say that the triangle ABC is equiangular with the triangle DEF, the angle ABC will be equal to the angle DEF, and the remaining angle, namely the angle at C, equal to the remaining angle, the angle at F. For, if the angle ABC is unequal to the angle DEF, one of them is greater. Let the angle ABC be greater ; and on the straight line AB, and at the point B on it, let the angle ABG be constructed equal to the angle DEF. [i. 33] Then, since the angle A is -equal to D, a .:. ui and the angle ABG to the angle DEF, therefore the remaining angle A GB is equal to the remaining angle DFE. [i. jj] Therefore the triangle ABG is equiangular with the triangle DEF. Therefore, as AB is to BG, so is DE to EF [vi. 4] But, as DE is to EF, so by hypothesis is AB to BC\ therefore AB has the same ratio to each of the straight lines BC, BG ; [v. 11] therefore BC is equal to BG, [v. 9] so that the angle at C is also equal to the angle BGC. [i. 5] VL 7] PROPOSITION 7 307 But, by hypothesis, the angle at C is less than a right angle ; therefore the angle BGC is also less than a right angle ; so that the angle A GB adjacent to it is greater than a right angle. [1. 13] And it was proved equal to the angle at F\ therefore the angle at F'\% also greater than a right angle. But it is by hypothesis less than a right angle : which is absurd. Therefore the angle ABC is not unequal to the angle DEF\ therefore it is equal to it. But the angle at A is also equal to the angle at D ; therefore the remaining angle at C is equal to the remaining angle at F, [i. 3a] Therefore the triangle ABC is equiangularwith the triangle DEF. But, again, let each of the angles at C, F be supposed not less than a right angle ; 1 say again that, in this case too, the triangle ABC is equiangular with the triangle DEF. For, with the same construction, we can prove similarly that BC is equal to BG \ so that the angle at C is also equal to the angle BGC. \y s] But the angle at C is not less than a right angle ; therefore neither is the angle BGC less than a right angle. Thus in the triangle BGC the two angles are not less than two right angles : which is impossible, [i- 17] Therefore, once more, the angle ABC is not unequal to the angle DEF; therefore it is equal to it. But the angle at A is also equal to the angle at D ; therefore the remaining angle at C is equal to the remaining angle at F, [i, 33} Mft^ BOOK VI [vi. 7 Therefore the triangle ABC is equiangular with the triangle DEF. Therefore etc. '■ . ■ ;- Q. E, D. Todhunter points out, after Walker, that some more words are necessary to make the enunciation precise : "If two triangles have one angle equal to one angle, the sides about other angles proportional <so that tht sides sttbttnding ihe equal anglts are homologous^. ..." This proposition is the extension to similar triangles of the ambiguous ^ase already mentioned as omitted by Euclid in relation to equality of triangles in all respects {cf. note following i. 26, Vol, 1. p. 306). The enunciation of vi. 7 has suggested the ordinary method of enunciating the ambiguous (ase where equality and not similarity is in question. Cf. Todhunter's note on 1. 26, Another possible way of presenting this proposition is given by Todhunter, The essential theorem to prove is : ff two triangles have two sides of the one proportional to two sides of the other, and the angles opposite to one pair of corresponding sides equal, the angles which are opposite to the other pair of €orresponding sides shell either ^e equal or lie together equal to two right angles. For the angles included by the proportional sides must be either equal or uneqtiaL If they are equal, then, since the triangles have two angles of the one equal to two angles of the other, respectively, they are equiangular to one another. We have therefore only to consider the case in which the angles included by the proportional sides are unequal. The proof is, except at the end, like that of vi. 7. Let the triangles ABC, DEF have the angle at A equal to the angle at D ; let AB be to BC as DE to EF, but let the angle ABCht not equal to the angle DEF. The angles ACB^ DFE shall be together equal to two right angles. For one of the angles ABC, DEF must be the greater. Let ABC\^ the greater; and make the angle ABG equal to the angle DEF Then we prove, as in vi. 7, that the triangles ABG, DEF are equiangular, whence AB is to BG as DE is to EF. But AS is to BC as DE is to EF, by hypothesis. Therefore BG is equal to BC, and the angle BGC is equal to the angle BCA. VI. 7. 8] PROPOSITIONS 7, 8 Now, since the triangles ABG, DMF'wfi equiangular, the angle SGA is equal to the angle EFD, Add to them respectively the equal angles BGC, EC A; therefore the angles BCA, EFD are together equal to the angles BGA, BGC, i.e. to two right angles. It follows therefore that the angles BCA, EFD must be either equal or supplementary. But (i), if each of them is less than a right angle, they cannot be supplementary, and they must therefore be equal ; (2) if each of them is greater than a right angle, Utey cannot be supplementary and must therefore be equal; (3) if one of them is a right angle, they are supplementary and also equal. Simson distinguishes the last case (3) in his enunciation : "then, if each of the remaining angles be either less or not less than a right angle, or if ont 0/ ihtm be a rigAt angle,., ," The change is right, on the principle of lestricring the conditions to the minimum necessary to enable the conclusion to be inferred. Simson adds a separate proof of the case in which one of the remaining angles is a right angle. " Lastly, let one of the angles at C, F, viz. the angle at C, be a right angle; in this case likewise the triangle ABC IS equiangular to the triangle DEF, For, if they be not equiangular, make, at the point B of the straight line AB, the ai^le ABG equal to the angle DEF\ then it may be proved, as in the first case, that BG vi equal vaBC. But the angle BCG is a right angle; therefore the angle BGC is also a right angle; whence two of the angles of the tri- angle BGC ^.Tc together not less than two right angles : which is impossible. Therefore the triangle ABC is equiangular to the triangle DEF." Proposition 8, . , 1/ in a rigkt-anghd triangle a perpendicular be drawn from ike right angle to the base, ike triangles adjoining ike perpendicular are similar both to the whole and to one another. Let ABC be a right-angled triangle having the angle BAC right, and let AD be drawn from A perpendicular to^C; I say that each of the triangles ABD, ADC is similar to the whole ABC and, further, they are similar to one another. ue BOOK VI [vi. 8 For, since the angle BAC is equal to the angle ADB, for each is right, and the angle at B is common to the two triangles ABC and ABD, therefore the remaining angle ACB is equal to the remaining angle BAD ; [.. 3^] therefore the triangle ABC is equi- angular with the triangle ABD. Therefore, as BC which subtends the right angle in the triangle ABC is to BA which subtends the right angle in the triangle ABD, ^o\% AB itself which subtends the angle at C in the triangle ABC to BD which subtends the equal angle BAD in the triangle ABD, and so also '\% AC to AD which subtends the angle at B common to the two triangles. [v,.4] Therefore the triangle ABC is both equiangular to the triangle ABD and has the sides about the equal angles proportional. Therefore the triangle ABC is similar to the triangle ABD. [VI. Def. i] Similarly we can prove that the triangle ABC is also similar to the triangle ADC ; therefore each of the triangles ABD, ADC is similar to the whole ABC. I say next that the triangles ABD, ADC are also similar to one another. For, since the right angle BDA is equal to the right angle ADC, and moreover the angle BAD was also proved equal to the angle at C, therefore the remaining angle at B is also equal to the remaining angle DAC; [»• 32] therefore the triangle ABD is equiangular with the triangle ADC. Therefore, as BD which subtends the angle BAD in the triangle ABD is to DA which subtends the angle at C in the triangle ADC equal to the angle BAD, so is AD itself which subtends the angle at B in the triangle ABD to DC which subtends the angle DAC in the triangle ADC equal VI. 8, g] PROPOSITIONS 8, 9 * jii to the angle at B, and so also is BA to AC, these sides subtending the right angles ; [vi. 4] therefore the triangle ABD is similar to the triangle ADC. [vi. Def. i] Therefore etc. PoRiSM. From this it is clear that, if in a right-angled triangle a perpendicular be drawn from the right angle to the base, the straight Hne so drawn is a mean proportional between the segments of the base. q. e. d. Sim son remarks on this proposition : "It seems plain that some editor has changed the demonstration that Euclid gave of this proposition : For, after he has demonstrated that the triangles are equiangular to one another, he particularly shows that their sides about the equal angles are proportionals, as if this had not been done in the demonstration of prop, 4 of this book : this superfluous part is not found in the translation from the Arabic, and is now left out." This seems a little hypercritical, for the "particular showing" that the sides about the e<jual angles are proportionals is really nothing more than a somewhat full citation of vi. 4. Moreover to shorten his proof still morci Simson says, after proving that each of the triangles ABD, ADC is similar to the whole triangle ABC, "And the triangles ABD, ADC being both equiangular and similar to ABC are equiangular and similar to one another," thus assuming a particular case of vi. zi, which might well be proved here, as EucSid proves it, with somewhat more detail. We observe that, here as generally, Euclid seems to disdain to give the reader such small help as might be afforded by arranging the letters used to denote the triangles so as to show the corresponding angular points in the same order for each pair of triangles ; A is the first letter throughout, and the other two for each triangle are in the order of the figure from left Co righL It may be in compensation for this that he states at such length which side corresponds to which when he comes to the proportions. In the Greek texts there is an addition to the Pori^m inserted after "(Being) what it iras required to prove," viz. "and further that between the base and any one of the segments the side adjacent to the s^ment is a mean proportional" Heiberg concludes that these words are an interpolation (i) because they come after the words wrcp iSh Stifm which as a rule follow the Porism, (2) they are absent from the best Theonine MSS., though P and Campanus have them without the wip {£« Stufiu. Heiberg's view seetns to be confirmed by the fact noted by Austin, that, whereas the first part of the Porism is quoted later in vi. 15, in the lemma before x. 33 and in the lemma after xin. 13, the second part \& prmted vci the former lemma, and elsewhere, as also in Pappus (in. p. 72, 9— »3). Proposition 9, From a given straight line to cut off a prescribed part. Let AB be the given straight line ; thus it is required to cut off from AB a prescribed part lis BOOK VI [«.» Let the third part be that prescribed. S Let a straight line AC he drawn through from A con- taining with AS any angle ; let a point /? be taken at random on AC, and let DB, EC be made equal to AD. [1- 3] 10 Let ^Cbe joined, and through D let DF be drawn parallel to it. [i- 31] Then, since FD has been drawn parallel to BC, one of the sides of the triangle ABC, therefore, proportionally, as CD is to DA, so is BFto FA. 1 • [n. a] •S But CD is double of DA ; therefore BF is also double of FA ; - therefore BA is triple of AF. Therefore from the given straight line AB the prescribed third part AF has been cut off. Q, E, F. 6. any angle. The exptessian here and in the two foltowing propositions is ruxoura yurtu, corresponding exactly to Tirj^ir (Ttj^miw which I have Iransfated u "« point (taken) at randffm"^ but *'an angle (talten) at random" would not be so appropriate where it is a question, not of taking any angle at all, but of drawing a straight line casuAlly so as to make any angle with another straight line. Simson observes that " this b demonstrated in a parti cuUt case, viz. that in which the third part of a straight line is required to be cut off; which b not at all like Euclid's inanner. Besides, the author of that demonstration, from four magnitudes being proportionals, concludes that the third of them is the same multiple of the fourth which the first is of the second ; now this is nowhere demonstrated in the sth book, as we now have it ; but the editor assumes it from the confused notion which the vulgar have of proportionals." The truth of the assumption referred to is proved by Simson in hb proposition D given above (p, laS); hence he is able to supply a general and legitimate proof of the present proposition. A " I^et AB be the given straight line ; it b required to cut oW any part from it. From the point A draw a straight line AC making any angle with AB; in AC take any point 2>, and take ^4 C the same multiple of AD that AB is of the part which is to be cut off from it ; join SC, and draw jD£ parallel to it ; then A Eh the part required to be cut off. VI. 9, lo] PROPOSITIONS 9, 10 "3 Because ED is parallel to one of the sides of the triangle ABC, \'a, to BC, as CD is to DA, so is BE to EA, [vi. 2] and, eomponaido, CA is to AD, as S A to AE. [v. 1 8] But CA is a multiple of AD ; therefore BA is the same multiple of AE. [Prop. D] Whatever part therefore AD is of AC, AE is the same part of AB ; wherefore from the straight line AB the part required is cut off." The use of Simson's Prop. D can be avoided, as noted by Camerer after Baermann, in the following way. We first prove, as above, that CA is to AD as BA is to AE. Then we infer that, alternately, CA is to BA as AD to AE. [v. 16] But AD is to AE as n . AD to n . AE {where n is the number of times that AD\% contained m AC); [v, 15] whence ACkIo AB as n . AD is to « . AE. [v, 1 1] In this proportion the first term is equal to the third ; therefore [v. 14] the second is equal to the fourth, so that AB is equal to n times AE. Prop. 9 is of course only a particular case of Prop 10. ' " Proposition id. To cut a given uncut straight line similarly to a given cut straight line. Let AB be the given uncut straight line, and AC the straight line cut at the points D, E ; and let them be so placed as to contain any angle ; let CB be joined, and through D, E let DF, EG be drawn parallel to BC, and through D let DHK be drawn parallel to AB. [1, 31] Therefore each of the figures FH, HB is a parallelogram ; therefore DH is equal to FG and HK to GB. [i. 34] Now, since the straight line HE has been drawn parallel to KC, one of the sides of the triangle DKC, therefore, proportionally, as CE is to ED, so is KH to HD. [vi. j] i«»4 "' BOOK VI [vi. 10, II ^ But KH is equal to BG, and HD to GF; ■ ' therefore, as C£ is to ED, so is 3G to GF, Again, since FD has been drawn parallel to GF, one ot the sides of the triangle j4 GF, therefore, proportionally, as FD is to DA, so is GF to FA. [vt. j] But it was also proved that, as CF is to FD, so is FG to GF; therefore, as CF is to FD, so is BG to GF, ^ and, as FD is to /?^, so is GF to /^^. Therefore the given uncut straight line AB has been cut similarly to the given cut straight line AC. Q. E. F. Proposition ii. ' - ' To two given straight lines to find a third proportional. Let BA, AC be the two given straight lines, and let them be placed so as to contain any angle ; thus it is required to find a third pro- portional to BA, AC. For let them be produced to the points D, E, and let BD be made equal to^C; [t. 3] let BC be joined, and through D let DF be drawn parallel to it. [i. 31] Since, then, BC has been drawn parallel to DF, one of the sides of the triangle ADE, proportionally, as AB is to BD, so is ^C to CF. [vi. aj But BD is equal to AC; therefore, as AB is to AC, so is AC tc CE. Therefore to two given straight lines AB, AC 3. third proportional to them, CF, has been found, Q. E. F. I. to And. The Greek word, bat and in tbe Bext two piopoHttom, is ir pnirivptir, liter*] I7 "to find in addilien." This proposition is again a particular case of the succeeding Prop, i », Given a ratio between straight lines, VI, ii enables us to find the ratio which i$ its duplicate;. vt. ii] PROPOSITIONS io-~u 1^ t. ,. Propos:tion 12. To three given straight lines to find a fourth proportional. Let A, B, C be the three given straight lines ; thus it is required to find a fourth proportional to A, B, C. B- C- Let two straight lines DE, DF be set out containing any angle EDF ; let DG be made equal to A, GE equal to B, and further DH equal to C\ let GH be joined, and let EF be drawn through E parallel to it. [1. 31] Since, then, GH has been drawn parallel to EF, one of the sides of the triangle DEF, therefore, as DG is to GE, so is DH to HF. [vi. a] But DG is equal to A, GE to B, and DH to C ; therefore, as A is to B, so is C to HF. Therefore to the three given straight lines A,B,Ca. fourth proportional HF has been found. Q. E..F. We have here the geometrical equivalent of the " rule of three." It is of course immaterial whether, as iti Euclid's proof, the first and second straight lines are measured on one of the lines forming the angle and the third on the other, or the first and third are measured on one and the second on the other. If it should be desired that the first and the required fourth be measured on one of the lines, and the second and third on the other, we can use the following construction. Measure -DE on one straight line equal to A, and on any other straight line making an angle with the first at the point D measure I>F equal to £, and DG equal to C, Join B.F, and through G draw GJf an/i-fiamiltl to EF, le. make the angle DGH equal to the angle DEF; let GH meet DE (produced if necessary) in H. »« BOOK VI [vi, I a— 14 DHis then the fourth proportional. For the triangles EDF, GDH are similar, and the sides about the equal angles are proportional, so that DE is to DFa& DG to DH, OT ^ is to ^ as C to DJf, Proposition 13. To two given straight lines to find a mean proportional. Let AB, BC be the two given straight lines ; thus it is required to find a mean proportional to AB, BC. Let them be placed in a straight line, and let the semicircle ADC be described or\ AC \ let BD be drawn from the point B at right angles to the straight line AC, , and let AD, DC be joined. Since the angle ADC is an angle in a semicircle, it is right. [ill. 31] And, since, in the right-angled triangle ADC, DB has been drawn from the right angle perpendicular to the base, therefore DB is a mean proportional between the segments of the base, AB, BC. [vi. 8, For.] Therefore to the two given straight lines AB^ BC a mean proportional DB has been found, Q. E. F. This proposition, the Book vi. version of ii. 14, is equivalent to the extraction of the square root. It further enables us, given a ratio between straight lines, to find the ratio which is its sub-dupiieate, or the ratio of which it is duplicate. , - Proposition 14, ', ^ In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional ; and equiangular parallelograms in which the sides about (he equal angles are reciprocally proportional are equal vj. 14] PROPOSITIONS 12— 1 4 at? ■^•^ Let AB, BC be equal and equiangular parallelograms having the angles at B equal, and let DB, BE be placed in a straight line ; therefore FB, BG are also in a straight line. [i. m] I say that, in AB, BC, the sides about the equal angles are reciprocally proportional, that is to say, that, as DB is to BE, so is GB to BF. For let the parallelogram FE be completed. Since, then, the parallelogram AB is equal to the parallelo- gram BC-, and FE is another area, therefore, as AB is to FE, so is BC to FE. [v. 7] But, as W^ is to FE, so is DB to BE, [vi, i] and, as BC is to FE, so is GB to BF, \id^ therefore also, as DB is to BE, so is GB to BF. [v. n] Therefore in the parallelograms AB, BC the sides about the equal angles are reciprocally proportional. ,^i Next, let GB be to BF as DB to BE-, .n I say that the parallelogram AB is equal to the parallelogram BC. For since, as DB Is to BE, so is GB to BF, while, as DB is to BE, so is the parallelogram AB to the parallelogram FE, [vi. i] and, as GB is to BF, so is the parallelogram BC to the parallelogram FE, [vi. i] therefore also, as AB is to FE, so is ^C to FE ; [v. n] therefore the parallelogram AB is equal to the parallelogram BC ... ^ ,. . . [V.9] Therefore etc. Q. E. D. De Morgan says upon this proposition : " Owing to the disjointed manner in which Euctid treats compound ratio, this prrvpositton is strangely out of place. It is a particular case of vi, 23, being that in which the ratio of the sides, compounded, gives a ratio of equality. The proper definition of four mapiitudes being reciprocally proportional is that the ratio compounded of thetr ratios is that of equality." 3i8 BOOK VI [vi. 14 It IS true that vi. 14 is a particular /case of vi. 23, but, if either is out of platt, it is rather the latter that should be placed before vi. 14, since most of the propositions between vi, 15 and vi. 23 depend upon vi. 14 and 15. But is perfectly consistent with Euclid's manner to give a particular case first and its extension later, and such an arrangement often has great advantages in that it enables the more difficult parts of a subject to be led up to more easily and gradually. Now, if De Morgan's view were here followed, we should, as it seems to me, be committing the mistake of explaining what is relatively easy to understand, viz, two ratios of which one is the inverse of the other, by a more complicated conception, that of compound ratio. In other words, it is easier for a learner to realise the relation indicated by the statement that the sides of equal and equiangular parallelograms are "recipro- cally proportior^al " than to form a conception of parallelograms such that " the ratio compounded of the ratio of their sides is one of equality." For this' reason I would adhere to Euclid's arrangement. The conclusion that, since I>B, BE are placed in a straight line, ES, BG are also in a straight line is referred to t. 14. The deduction is made clearer by the following steps. The angle DBF'm equal to the angle GBE; " add to each the angle FB£ ; therefore the angles DBF, FBE are together equal to the angles GBE, FBE. {C. N. i] But the angles DBF, FBE are together equal to two right angles, [i. 13] therefore the angles GBE, FBE are together equal to two right angles, [C.N.t] and hence FB, BG are in one straight line. [i. 14] The result is also obvious from the converse of 1. 15 given by Proclus (see note on i. 15). The proposition vi. 14 contains a theorem and one partial converse of it; so also does vi. 1 5. To each proposition may be added the other partial converse, which may be enunciated as follows, the words in square brackets applying to the case of triangles (vi. 15). Equal paralhlogrami \triattgUs\ which havt the sides absut one angle in each redprocally proportional art equiansular \have the angles included by those sides either equal or supplementary^ Let ABf BC be equal parallelograms, or let FBD, EBG be equal ■n. 14, isl PROPOSITIONS 14, 15 119 triangles, such that the sides about the angles at B are reciprocally propor- tional, i.e. such that DB : BE = GB : BF. We shall prove that the angles FED, EBG are either equal or supple- mentary. Place the figures so that DB. BE are in one straight line. Then FB, BG are either in a straight line, or not in a straight line. (i) If FB, BG are in a straight line, the figure of the proposition (with the diagonals FD, EG drawn) represents the facts, and the angle FBD is equal to the angle EBG. ['. 15] (2) If J'B, BG are not in a straight line, produce FB to H so that BH may be equal to BG. Join EJf, and complete the parallelogram EBHK. Now, since DB : BE ^ GB \ BF xbA. GB = HB, ~ DB : BE = HB . BF, and therefore, by vi. r4 or 15, the parallelograms j4B, BK 3.x^ equal, or the triangles FBD, EB/fare equal. But the parallelograms AB, BCart eciual, and the triangles FBD, EBG are equal ; therefore the parallelograms BC, BK are equal, and the triangles EBH, EBG are equal. Therefore these parallelograms or triangles are within the same parallels : that is, G, C, H, K are in a straight line which is parallel to DE. [1, 39] Now, since BG, BHare. equal, the angles BGH, BHG are equal. By parallels, it follows that the angle EBG is equa' to the angle DBH, whence the angle EBG is supplementary to the angle FBD. Proposition 15. In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional ; and those triangles which have otie angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal Let ABC, ADE be equal triangles having one angle equal to one angle, namely the angle BAC to the angle DAE, I say that in the triangles ABC, ADE the sides about the equal angles are reciprocally proportional, that is to say, that, as CA is to AD, so is EA to AB, aao BOOK VI [vi. 15 For let them be placed so that CA is in a straight line with AD; therefore EA is also in a straight line with AB. [i. 14] Let BD be joined. Since then the triangle ABC is equal to the triangle ADE, and BAD is another area, therefore, as the triangle CAB is to the triangle BAD, so is the triangle EAD to the triangle BAD. [v. 7] But, as CAB is to BAD, so is CA to AD, [vi. i] and, as EAD is to BAD, so is EA to AB. \id:\ Therefore also, as CA is to AD, so is EA to AB. (v. n] Therefore in the triangles ABC, ADE the sides about the equal angles are reciprocally proportional. Next, let the sides of the triangles ABC, ADE be reci- procally proportional, that is to say, let EA be to AB as CA to AD ; I say that the triangle ABC is equal to the triangle ADE. For, if BD be again joined, 1 , since, as CA is to AD, so is EA to AB, while, as CA is to AD, so is the triangle ABC to the triangle BAD, and, as EA is to AB, so is the triangle EAD to the triangle BAD, [vt. ,] therefore, as the triangle ABC is to the triangle BAD, so is the triangle EAD to the triangle BAD. [v. 11] Therefore each of the triangles ABC, EAD has the same ratio to BAD. Therefore the triangle ABC is equal to the triangle EAD. [v. 9] Therefore etc. ■ -> ;■ r Q. E. D. As indicated in the partial converse given in the last note, this proposition is equally true if the angle included by the two sides in one triangle h supplementary, instead of being equal, to the angle included by the two sides in the other. VI. 15, 1 6] PROPOSITIONS 15, i« »ai Let ABC, ADE be two tria.ngles such that the angles BA C, DAE are supplementary, and also .^ CA:AD = EA: AB. In this case we can place the triangles so that CA is in a straight line with AD, and AB lies along AE (since the angle EAC, being supple- mentary to the angle EAD, is equal to the anglo BAC). If we join BD, the proof given by Euclid applies to this case also. It is true that vi, 15 can be immediately inferred from vi. 14, since a triangle is half of a parallelogram vrith the same base and height. But, Euclid's object being to give the student a grasp of mttkods rather than results, there seems to be no advantage in deducing one proposition from the other instead of using the same method on each. Proposition 16. If four straight lines be proportional, the rectangle con- tained by the extremes is equal to the rectangle contained by the means ; and, if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines will be proportional. Let the four straight lines AB, CD, E, F be propoitional, so that, as AB is to CD, so is B to F; I say that the rectangle contained by AB, F is equal to the rectangle contained by CD, E. Let AG, CH be drawn from the points A, C ^t right angles to the straight lines AB, CD, and let AG he made equal to F, and Cff equal to E. Let the parallelograms BG, DH be completed. Then since, as AB Is to CD, so is .£■ to F, while E is equal to CH, and f^ to AG, therefore, as AB is to CD, so is CH to AG. Therefore in the parallelograms BG, DH the sides about the equal angles are reciprocally proportional. 3«3 BOOK VI . , ■ [VI. 1 6 But those equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal ; [VI. 14] therefore the parallelogram BG is equal to the parallelogram And BG is the rectangle AB, F, for AG is equal to F; and DH is the rectangle CD, E, for ^ is equal to CH \ therefore the rectangle contained by AB, F is equal to the rectangle contained by CD, E. Next, let the rectangle contained by AB, F be equal to the rectangle contained by CD, E ; I say that the four straight lines will be proportional, so that, as AB is to CD, so is E to F. For, with the same construction, since the rectangle AB, /^ is equal to the rectangle CD, E, and the rectangle AB, F is BG, for AG is equal to F, and the rectangle CD, £ is D/f, for C// is equal to £, therefore BG is equal to D/f. And they are equiangular , •, •• ■ . But in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional. [vi. 14] Therefore, as AB is to CD, so is Cff to A G. But CH is equal to E, and AG to F; therefore, as AB is to CD, so is £" to F. Therefore etc. q. e. d. Thb proposition is a, particular case of vi. 14, but one which is on all accounts worth separate statement. It may also be enuticiated in the follow- ing form : Jitetartgki which have thtir bases rtdprocaUy proporiumal to ihcir Atighis- art equcd in arta; and egml rectangles have their bases redproealfy proportional to thdr heights. Since any fkarallelogTam is equal to a rectangle of the same height and Dn the same base, and any triangle with the same height and on the same base is equal to half the parallelc^ram or rectangle, it follows that Equal parallelograms or triangles have their bases reciprocally proportional to their heights and vice vena. The present place is suitable for giving certain important propositions, including those which Simson adds to Book vi. as Props, B, C and D, which are proved directly by means of vi, 16. I. Proposition B is a particular case of the following theorem. J/ a circle be circumscribed about a triangle ABC and there be drawn through A any two straight lines either both within or both without the an^ BAG, pm. VI. 1 6] PROPOSITION r6 '«t3 AD mteting BC {produad if Mcttsary) in D and AE mealing the circle again in E, suth that the angles DAB, EAC are equal, then the rectangle AD, AE is equal to the rectangle BA, AC. Join CE. The angles BAD, £AC 3,k equal, by hypothesis ; and the angles ABZ>, AEC are equal, [in. n, as] Therefore the triangles ABD, AEC are equiangular. Hence BA is to AD as EA is to AC, and therefore the rectangle BA, AC is equal to the rectangle AD, AE, [VI. 1 6] There are now two particular cases to be considered. (a) Suppose that AD, AE coincide ; ADE will then bisect the angle BAC. ' • (b) Suppose that AD, v4£ are in one straight line but that D, E ate on opposite sides of A ; AD will then bisect the external angle at A. e In the fiist case {a) we have a; >•' - < the rectangle BA, ^iC equal to the rectangle EA, AD; and the rectangle EA, AD is equal to the rectangle ED, DA together with the square on AD, [\i. 3] i.e. to the rectangle BD, DC together with the square on AD. [11 1. 35] Therefore the rectangle BA, AC b, equal to the rectangle BD, DC tf^etber with the square on AD. [This is Simson's Prop B] In case (^) the rectangle EA, AD is equal to the excess of the rectangle ED, DA over the square on AD ; therefore the rectangle BA, AC k equal to the excess of the rectan^e BD, DC over the square on AD. a$4 BOOK VI [vt. i6 The following converse of Simson's Prop. B may be given : ^ a stra^ht Uiu AD ^ drawn from the virtex K of a Iriat^lt to meH the base, S9 that the square on AD together with the recian^e BD, DC is tqual to the rectangU BA, AC, the line AD will biieet the angle BAG exapt when the sides AB, AC are tqual, in which atse every line drawn to the base will have the property men- timed, Let the circumscribed circle be drawn, and let AD produced meet it in E; join CE. The rectangle S£>, DC is equal to the rectangle ED, DA. [iii. 35] Add to each the square on AD ; therefore the rectangle BA, AC\s equal to the rectangle EA, AD. [hyp. and Ii. 3] Hence AS is to AD as AE to AC. [vi, 16] But the angle ABD is equal to the angle AEC. [111. 11] Therefore the angles BDA, EC A are either equal or supplementary. [vi. 7 and note] (o) If they are equal, the angles BAD, SAC are also equal, and AD bisects the angle BAC. {b) If they are supplementary, the angle ADC must be equal to the angle ACE. Therefore the angles BAD, ABD are together equal to the angles ACB, BCE, i.e. to the angles A CD, BAD. Take away the common angle BAD, and the angles ABD, ACD are equal, or jJJ is equal to AC. Euclid himself assumes, in Prop. 67 of the Data, the result of so much of this proposition as relates to the case where BA = AC. He assumes namely, without proof, that, if BA -^AC, and if ZJ be any point on BC, the rectangle BD, Z>C together with the square on AD\^ equal to the square on AB. Proposition C. Jffrom any angle of a triangle a straight line be drawn perpendicular to the opposite side, the rectangle contained by the other two sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle (iraimscriied about the triangle. Let .i4^C be a triangle and AD the perpendicular on AB. Draw the diameter A£ of the circle circumscribed about the triangle ABC. Then shall the rectangle BA, ACh^ equal to the rectangle EA, AD. Join EC. VI. 1 6] PROPOSITION 1 6 "5 Since the right angle BDA is equal to the right angle ECA in a semi- circle, [ill. 31] and the angles ABD, AEC in the same s^pnent are equal, [in. 21] the triangles ABD, AEC are equiangular. Therefore, as ^^4 is to AD, so is EA to AC, [vj. 4] whence the rectangle BA, AC is equal to the rectangle EA, AD. [vi, 16] This result corresponds to the trigonometrical formula, for JP, the radius of the circumscribed circle, E = 4& Proposition D. This is the highly important lemma given by Ptolemy (ed, Heiberg, Vol. i, pp. 36 — 7) which is the basis of his calculation of the table of chords in the section of Book i. of (he /xcyoAi; trvrraii% entitled " concerning the siite of the straight lines [i.e. chords] in the circle " (rtpl i-ijt TnjKiKonfTot too' ^v riji KixXif The theorem may be enunciated thus. 7%e ruiangk wfitaimd by iht diagsnah of any quadrilateral inscribtdin a drclt is equal to the sum afthe nclangks contained by the pairs of opposite sides. I shall give the proof in Ptolemy's words, with the addition only, in brackets, of two words applying to a second figure not giver by Ptolemy. " Let there be a circle with any quadrilateral ABCD inscribed in it, and \exAC, BD be joined It is to be proved that the rectangle contained by ACznA BD is equal to the sum of the rectangles AB, DC and AD, BC. For let the angle ABE be made equal to the angle contained by DB, BC. -i' (jMl If then we add [or subtract] the angle EBD, ' ~ < the angle ABD will also be equal to the angle ESC. But the angle BDA is also equal to the angle BCE, ' [m. ai] for they subtend the same segment ; therefore the triangle ABD is equiangular with the triangle EBC. Hence, proportionally, as BC is to CE, so is BD to DA. [vi. 4] Therefore the rectangle BC,ADk equal to the rectangle BD, CE. [VI. 16] Again, since the angle ABE is equal to the angle DBC, and the angle BAE is also equal to the angle BDC, [m. ai) the triangle ABE is equiangular with the triangle DBC. aa6 BOOK VI [vi. 1 6 [VI. 4] [vi. i6] Therefore, proportionallj', as £A is to A£, so is SJD to DC; therefore the rectangle BA, DC is equal to the rectangle BD, AE. But it was also proved that the rectangle BC, AD is equal to the rectangle BD^ CE\ therefore the rectangle AC, BD as a whole is equal to the sum of the rectangles AB, i»Cand AD, BC: (being) what it was required to prove." Another proof of this proposition, and of its converse, is indicated \yj Dr Lachlan (Elements of Euclid, pp. 273 — 4). It depends on two preliminary propositions. (1) If two tirdes bi divided, by a chord in each, into segments which are similar respeettvcfy, the chords are proportional to the corresponding diameters. The proof is instantaneous if we join the ends of each chord to the centre of the circle which it divides, when we obtain two similar triangles. {1) IfYibe any point on the circle circumscribed about a triangle ABC, and DX, DY, DZ be perpendicular to the sides BC, CA, AB of the triangle respectively, then X, Y, Z lie in one straight line ; and, conversely, if the feet of the perpendiculars from any point D on the sides of a triangle lie in one straight line, D ties on the circle circumscribed about the triangle. The proof depending on ill. 21, 22 is well known. Now suppose that D is any point in the plane of a triangle ABC, and that DX, D y, DZ are perpendicular to the sides BC, CA, AB respectively. Join YZ, DA. Then, since the angles at ]^ Z are right, A, Y, D, Z lie on a circle of which DA is the diameter. And YZ divides this circle into segments which are similar respectively to the segments into which BC divides the circle circumscribing ABC, since the angles ZAY, BAC coincide, and their supple- ments are equal. Therefore, if i^ be the diameter of the circle circumscribing ABC, BChlodas YZistoDA; and therefore the rectangle AD, BC is equal to the rectangle d, YZ. Similariy the rettangle BD, CA is equal to the rectangle d, ZX, and the rectangle CD, AB\& equal to the rectangle d, XY. Hence, in a quadrilateral in general, the rectangle contained by the diagonals is less than the sum of the rectangles contained by the pairs of opposite sides. Next, suppose that D lies on the circle circum- scribed about ABC, but so that A, B, C, D follow each other on the circle in this order, as in the figure annexed. Let DX, DY, DZ be perpendicular to BC, CA, AB respectively, so that X, Y, Zare in a straight line. Then, since the rectangles AD, BC; BD, CA; CD, AB are equal to the rectangles d, YZ; d, ZX; d, .VF respectively, and XZi% equal to the sum of VI. i6] PROPOSITION id aaf Xy, YZ, so that the rectangle 4 XZ is equal to the sum of the rectangles 4 JfKand d, YZ, It follows that the rectangle AC, BD is equal to the sum of the rectangles AD, BCand AB, CD. Cenvirsefy, if the latter statement is true, while we are supposed to know nothing about the position of D, it follows that XZ must be equal to the sum of XY, YZ, so that X, Y, Z must be in a straight line. Hence, from the theorem (2) above, it follows that D must lie on the circle circumscribed about ABC, i.e. that A BCD is a quadrilateral about which a circle can be described. All the above propositions can be proved on the basis of Book iii. and without using Book vi., since it is possible by the aid of 111.^21 and 35 alone to prove that in equiaitgular triangles the rectangles contained by the iwh- correspsnding sides about equal angles are equal to one another (a result arrived at by combining vi, 4 and v[. 16). This is the method adopted by Casey, H. M. Taylor, and Lachlan ; but I fail to see any particular advantage in it Lastly, the following proposition may be given which Playfair added as VI. E. It appears in the Data of Euclid, Prop. 93, and may be thus enunciated. If the angle BAG of a triangle ABC be bisected by the straight line AD meeting (he tirde circumscribed about the triangle in D, and if BD be joined, then the sum of BA, AC /> to AD as BC is to BD, Join CD. Then, since AD bisects the angle BAC, the subtended arcs SD, DC, and therefore the chords BD, DC, are equal. (i) The result can now be easily deduced from Ptolemy's theorem. For the rectangle AD, BC is equal to the sum of the rectangles AB, DC and .4C, BD, i.e. {since BD, CD are equal) to the ret tangle contained by BA + ACmdBD. Therefore the sum of BA, AC is to AD as BC is to BD. [vi. 16] (2) Euclid proves it differently in Data, Prop. 93, Let AD meet BC in JB. Then, since A£ bisects the angle BAC, BA i& to AC as BE to £C, [vi. 3] or, alternately, AB is to BE as AC to C£. [v. r6] Therefore also BA + AC is to BC BS AC to CE. [v, la] Again, since the angles BAD, EA C are equal, and the angles ADB, ACE are also equal, [in. 2i] the triangles ABD, A EC are equiangular. Therefore ^C is to CE as AD to BD. [vi, 4] ail BOOK VI [vi. 16, 17 Hence BA^ACisXaBCiBADloBD, [v. 11] and, alternately, BA + AC is to AD as BC is to BD. (v. 16] Euclid concludes that, if the circle ABC is pven in magnitude, and the chord BC cuts off a segment of it containing a given angle (so that, by Data Prop. 87, 2fCand also BD are given in magnitude), , ,, ,^^ the ratio of BA + AC to AD is given, and further that (since, by similar triangles, BD is to D£ as ^4 C is to C£, vhi\t BA + AC is to BCsi&ACis to C£), the rectangle (BA + AC), D£, being equal to the rectangle BC, BD, is also given. Proposition 17 // three straight lines be proportional, the rectangle con- tained by the extremes is equal to the square on the mean; and, if the rectangle contained by the extremes be equal to the square on the mean, the three straight lines will be proportional. Let the three straight lines A, B, C be proportional, so that, as A is to B, so is £ to C ; I say that the rectangle contained by A, C is equal to the square on B. Let D be made equal to B. Then, since, as A is to B, so is ^ to C, and B is equal to D, therefore, as A is to B, so is Z* to C, But, if four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means. [vi, i6' Therefore the rectangle A, C h equal to the rectangle B,D. But the rectangle S, D is the square on B, for B is equal to D ; therefore the rectangle contained by A, C is equal to the square on B. Next, let the rectangle A, Che equal to me square on B I say that, as ^ is to B, so is B to C. VI. 17, i8] PROPOSITIONS 16—18 aaff For, with the same construction, since the rectangle A, C is equal to the square on B, while the square on B is the rectangle B, D, for B is equal to A therefore the rectangle A, C '\s equal to the rectangle B, D. But, if the rectangle contained by the extremes be equal to that contained by the means, the four straight lines are proportional. [vi. 16] Therefore, as A is to B, so is D to C But 5 is equal to /?; '•'' therefore, as A is to B, so is B to C. Therefore etc. q. e. d. VI. 17 is, of course, a particular case of vi. 16. Proposition 18. On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure. Let AB be the given straight line and CE the given rectilineal figure ; thus it is required to describe on the straight line AB a rectilineal figure similar and similarly situated to the recti- lineal figure CE. Let DF be joined, and on the straight line AB, and at the points A, B on it, let the angle GAB be constructed equal to the angle at C, and the angle ABG equal to the angle CDF. [1. 13] Therefore the remaining angle CFD is equal to the angle AGB ; [I. 32] therefore the triangle FCD is equiangular with the triangle GAB. Therefore, proportionally, as FD is to GB, so is FC to GA, and CD to AB. «39 BOOK VI [VL 18 Again, on the straight line BG, and at the points B, G on it, let the angle BGH be constructed equal to the angle DFE, and the angle GBH equal to the angle FDE. [i. »3] Therefore the remaining angle at E is equal to the re- maining angle at H ; [i. 3^] therefore the triangle FDE is equiangular with the triangle GBH\ therefore, proportionally, as FD is to GB, so is FE to GH, and ED to HB. [vi. 4] But it was also proved that, as FD is to GB^ so is FC to GA, and CD to AB ; therefore also, as FC is to AG, so is CD to AB, and /^£' to G//, and further £"/? to HB. And, since the angle CFD is equal to the angle AGB, and the angle DFE to the angle ^G^//, therefore the whole angle CFE is equal to the whole angle AGH. For the same reason the angle CDE is also equal to the angle ABH. And the angle at C is also equal to the angle at A, and the angle at E to the angle at H. Therefore AH is equiangular with CE ; and they have the sides about their equal angles proportional ; therefore the rectilineal figure AH is similar to the rectilineal figure CE. [vi. Def. i] Therefore on the given straight line AB the rectilineal figure AH has been described similar and similarly situated to the given rectilineal figure CE. .. Q. E. F. Simson thinks the proof of this proposition has been vitiated, his grounds for this view being (1) that it is demonstrated only with reference to quadrilaterals, and does not show how tt may be extended to figures of five or more sides, (i) that Euclid infers, from the fact of two triangles being equiangular, that a side of the one is to the corresponding side of the other as another side of the first is to the side corresponding to it in the other, i.e. he permutes, without mentioning the fact that he does so, the proportions obtained in vi. 4, whereas the proof of the very next proposition gives, in a similar case, the intermediate step of permutation. I think this is hyper- criticism. As regards (a) it should be noted that the permuted form of the proportion is arrived at first in the proof of vi, 4 ; and the omission of the VI. 1 8] PROPOSITION i8 '40- intermediate step of allertiandOy whether accidental or not, is of no importance. On the other hand, the use of this form of the proportion certainly simplifies the proof of the proposition, since it makes unnecessary the subsequent IX aequali steps of Simson's proof, their place being taken by the inference [v. 1 1] that ratios which are the same with a third ratio are the same with one another. Nor is the first objection of any importance. We have only to take as the given polygon a polygon of five sides at least, as CDEFG, pin one extremity of CD, say D, to each of the angular points other than C and E, and then use the same mode of construction as Euclid's for any number of successive triangles as ABL, LBK, etc, that may have to be made. Euclid's con- struction and proof for a quadrilateral are quite sufficient to show how to deal with the case of a figure of five or any greater number of sides. Clavius has a construction which, given the power of moving a figure bodily from one position to any other, is easier. CDEFG being the given polygon, join CE, CF. Place AB on CD so that A falls on C, and let B fall on ly, which may either lie on CD or on CD produced. Now draw DE parallel to DE, meeting CE, produced if necessary, in E, EF' parallel to EF, meeting CF, produced if necessary, in E", and so on. Let the parallel to the last side but one, FG^ meet CG, produced if necessary, in C. Then CUE FG is similar and similarly situated to CDEFG, and it is constructed on CD, a straight Hne equal to AB. The proof of this is obvious. A more general construction is indicated in the subjoined figure. If CDEFG be the given polygon, suppose its angular points all joined to any point O and the connecting straight lines produced both ways. Then, if CD, a straight line equal io AB, be placed so that it is parallel to CD, and C, D lie respectively on OC, OD (this can of course be done by finding fourth proportionals), we have only to draw UE, EF, etc., parallel to the corresponding sides of the original polygon in the manner shown. a$a BOOK VI [VL i8, 19 De Morgan would rearrange Props. 18 and 20 in the following manner. He would combine Prop. 18 and the first part of Prop, ao into one, with the enunciation: E \FJ.S^-.:ifi-'*^^-.'.-~-----^ Pairs of similar triangits, similarly put tcgeihtr, give similar figures ; and every pair of similar figures is composed of pairs of similar triangles similarly put together. He would then make ihs problem of vi. 18 an application of the first part In form this would certainly appear to be an improvement; but, provided that the relation of the propositiotis is understood, the matter of form is perhaps not of great importance. Proposition 19. Similar triangles are to one another in the duplicate ratio of ike corresponding sides. Let ABC, DEFhe. similar triangles having the angle at B equal to the angle at £, and such that, as AB is to BC, so s is DE to EF, so that BC corresponds to EF; [v, Def. n] I say that the triangle ABC has to the triangle DEF a ratio duplicate of that which BC has to EF. For let a third proportional BG be taken to BC, EF, so that, as BC is to EF, so is EF to BG ; [vi. 1 1] 10 and let AG be joined. Since then, as AB is to BC, so is DE to EF, therefore, alternately, as AB is to DE, so is BC to EF. [v. 16] VI, 19] PROPOSITIONS t8, 19 S33 But, as BC is to EF, so is EF to BG ; ■" therefore also, as AB is to DE, so is EF to BG. [v. n] IS Therefore in the triangles ABG, DBF the sides about the equal angles are reciprocally proportional. But those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal; [vi. 15] 30 therefore the triangle ABG is equal to the triangle DEF. Now since, as BC is to EF, so is EF to BG, and, if three straight lines be proportional, the first has to the third a ratio duplicate of that which it has to the second, [v, Def. 9] therefore BC has to BG a ratio duplicate of that which CB as has to EF. But, as CB is to BG, so is the triangle ABC to the triangle ABG ; [vi. i) therefore the triangle ABC also has to the triangle ABG a ratio duplicate of that which BC has to EF. 30 But the triangle ABG is equal to the triangle DEF; therefore the triangle ABC also has to the triangle DEF a ratio duplicate of that which BC has to EF. ...... ,. ,. , Therefore etc. •I PoRiSM. From this it is manifest that, if three straight 35 lines be proportional, then, as the first is to the third, so is the figure described on the first to that which is similar and similarly described on the second, Q. E. D. 4. uid such that, as AB is \Q BC, so Is DE to BP, Utcrally "(tiiangls) having the anjfle at B equal to the anj;1e at f , and (AavtHg). as AB to BC^ so DK to EF^ Having combined Prop. [8 and the first part of Prop. 20 as just indicated, De Morgan would tack on to Prop. 19 the second part of Prop. 20, which asserts that, if similar polygons be divided into the same number of similar triangles, the triangles are " hemokgoui to the wholes " (in the sense that the polygons have the same ratio as the corresponding triangles have), and that the polygons are to one another in the duplicate ratio of corresponding sides. This again, though no doubt an improvement of form, would necessitate the drawing over again of the figure of the altered Proposition 18 and a certain amount of repetition. Agreeably to his su^estion that Prop, 23 should come before Prop. 14 which is a particular case of it, De Morgan would prove Prop. 19 for parctUtlsgrams by means of Prop, 13, and thence infer the truth of it for '*^ BOOK VI [vi. 19 triangles or the halves of the parallelograms. He adds ; " The method of Euclid is an elegant application of the operation requisite to compound equal ratios, by Which the conception of the process is lost sight of." For the general reason given in the note on vi, 14 above, I think that Euclid showed the sounder discretion in the arrangement which he adopted. Moreover it is not easy to see how performing the actual operation of compoutiding two equal ratios can obscure the process, or the fact that two equal ratios are being comfXJunded. On the definition of compounded ratios and duplicate ratio, De Morgan has himself acutely pointed out that "composition" is here used for the process of detecting the single alteration which produces the effect of two or more, the duplicate ratio being the result of compounding two equal ratios. The proof of vi. 19 does in fact exhibit the single alteration which produces the effect of two. And the operation was of the essence of the Cireek geometry, because it was the manipulation of ratios in this manner, by simplification and transformation, that gave it so much power, as every one knows who has read, say, Archimedes or ApoUonius. Hence the introduction of the necessary operation, as well as the theoretical proof, in this proposition seems to me to have been distinctly worth while, and, as it is somewhat simpler in this case than in the more general case of vj. 23, it was in accordance with the plan of enabling the difficulties of Book vi. to be more easily and gradually surmounted to give the simpler case first. That Euclid wished to emphasise the importance of the method adopted, as well as of the result obtained, in vi. 19 seems to me clearly indicated by the Porism which follows the proposition. It is as if he should say : "I have shown you that similar triangles are to one another in the duplicate ratio of corresponding sides; but I have also shown you incidentally how it is possible to work conveniently with duplicate ratios, viz. by transforming them into simple ratios between straight lines. I shall have occasion to illustrate the use of this method in the proof of vi. 22." Tlie Porism to VI. 19 presents one difficulty. It will be observed that it speaks of ^^ figure ((Bos) described on the first straight line and of that which is similar and similarly described on the second. If " figure " could be regarded as loosely used for the figure of the proposition, i.e. for a triangle, there would be no difficuity. If on the other hand " the figure " means any rectilineal figure, i.e. any p>olygon, the Porism is not really established until the next proposition, vi, 20, has been proved, and therefore it is out of place here. Yet the correction Tpiynivoi', triangle, for Ahot, figure, is due to Theon alone ; P and Campanus have " figure," and the reading of Philoponus and Psellus, TtTpayiiivov, square, partly supports (TS<n, since it can be reconciled with t'Sot but not with Tpiyuvoi'. Again the second Porism to vj. jo, in which this Porism is reasserted for any rectilineal figure^ and which is omitted by Campanus and only given by P in the margin, was probably interpolated by Theon. Heiberg concludes that Euclid wrote "figure" {<ISot), and Theon, seeing the difficulty, changed the word into " triangle " here and added For. 2 to VI. «o in order to make the matter clear. If one may hazard a guess as to how Euclid made the slip, may it be that he first put it after vi 20 and then, observing that the expression of the duplicate ratio by a single ratio between two straight lines does not come in vi. so but in vi. rg, moved the Porism to the end of vi. 19 in order to make the connexion clearer, without noticing that, if this were done, tTSo! would need correction ? The following explanation at the end of the Porism is bracketed by Heiberg, viz. "Since it was proved that, as CB is to BG, so is the triangle vt. 19, 20] PROPOSITIONS 19, 20 23s ABCio the triangle ABG, that is DEF." Such explanations in Porisms are not in Euclid's manner, and the words are tiot in Campanus, though they date from a time earlier than Theon. Proposition 20. Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding 5 side. Let ABCDE, FGHKL be similar polygons, and let AB correspond to FG ; I say that the polygons ABCDE, FGHKL are divided into similar triangles, and into triangles equal in multitude and in 10 the same ratio as the wholes, and the polygon ABCDE has to the polygon FGHKL a ratio duplicate of that which AB has to FG. Let BE, EC, GL, Z^ be joined. f.h Now, since the polygon ABCDE is similar to the polygon ,5 FGHKL, the angle BAE is equal to the angle GFL ; and, as BA is to AE, so is GF to FL, [vi. Def. i] Since then ABE, FGL are two triangles having one angle equal to one angle and the sides about the equal angles K proportional, therefore the triangle ABE is equiangular with the triangle FGL ; [v.. 6] so that it is also similar ; [vi. 4 and Def. i] therefore the angle ABE is equal to the angle FGL. 136 ■■•- BOOK VI :. [vt. 20 »S But the whole angle ABC is also equal to the whole angle FGH because of the similarity of the polygons ; therefore the remaining angle EBC is equal to the angle L,GH. , And, since, because of the similarity of the triangles ABE, 3PFGL, as EB is to BA, so is LG to GF, and moreover also, because of the similarity of the polygons, as AB is to BC, so is FG to GH, therefore, ex aequali, as EB is to BC, so is Z G^ to GH ; [v. aa] 35 that is, the sides about the equal angles EBC, LGH are proportional ; therefore the triangle EBC is equiangular with the triangle LGH, [vi. 6] so that the triangle EBC is also similar to the triangle ¥>LGH. [vi. 4 and Def. i] For the same reason the triangle ECD is also similar to the triangle LHK. Therefore the similar polygons ABCDE, FGHKL have been divided into similar triangles, and into triangles equal in 45 multitude. I say that they are also in the same ratio as the wholes, that is, in such manner that the triangles are proportional, and ABE, EBC, ECD are antecedents, while FGL, LGH, LHK are their consequents, and that the polygon ABCDE so has to the polygon FGHKL a ratio duplicate of that which the corresponding side has to the corresponding side, that is AB to FG. For let AC, FH be joined. Then since, because of the similarity of the polygons, iS the angle ABC is equal to the angle FGH, and, as AB is to BC, so is FG to GH, the triangle ABC is equiangular with the triangle FGH ; [VI. 6] therefore the angle BAC is equal to the angle GFH, and the angle BCA to the angle GHF. 60 And, since the angle BAM is equal to the angle GFN, and the angle ABM is also equal to the angle FGN, VI. jo] proposition lo 83J therefore the remaining angle A MB is also equal to the remaining angle FNG ; [i. 3a] therefore the triangle ABM is equiangular with the triangle 6s FGN. Similarly we can prove that the triangle BMC is also equiangular with the triangle GNH, Therefore, proportionally, as AM is to MB, so is FN to NG, 70 and, as BM is to MC, so is GN to NH ; so that, in addition, ex aequali, as AM is to MC, so is FN to NH. But, as AM is to MC, so is the triangle ABM to MBC, and AME to EMC; for they are to one another as their 7S bases. [vi. i] Therefore also, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents ; [v. is] therefore, as the triangle A MB is to BMC, so is ABE to CBE. 80 But, as AMB is to BMC, so is AM to MC ; therefore also, as AM is to MC, so is the triangle ABE to the triangle EBC For the same reason also, as FN is to NH, so is the triangle FGL to the triangle 8s GLH. And, as AM is to MC, so is FN to NH; therefore also, as the triangle ABE is to the triangle BEC, so is the triangle FGL to the triangle GLH ; and, alternately, as the triangle ABE is to the triangle FGL, 90 so is the triangle BEC to the triangle GLH. Similarly we can prove, if BD, GK be joined, that, as the triangle BEC is to the triangle LGH, so also is the triangle ECD to the triangle LHK. And since, as the triangle ABE is to the triangle FGL, 9S so is EBC to LGH, and further ECD to LHK, therefore also, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents ; [v. 13 therefore, as the triangle ABE is to the triangle FGL, so is the polygon ABCDE to the polygon FGHKL. i3S BOOK VI [vi. 10 too But the triangle ABE has to the triangle FGL a ratio duplicate of that which the corresponding side AB has to the corresponding side FG\ for similar triangles are In the duplicate ratio of the corresponding sides. [vi. 19] Therefore the polygon ABCDE also has to the polygon ■OS FGHKL a ratio dupUcate of that which the corresponding side AB has to the corresponding side FG. Therefore etc. PoRiSM, Similarly also it can be proved in the case of quadrilaterals that they are in the duplicate ratio of the 110 corresponding sides. And it was also proved in the case of triangles; therefore also, generally, similar rectilineal figures are to one another in the duplicate ratio of the corresponding sides. Q. E, D. J. in the same ratio as the wholes. The Mune word J/iiXnTmi is used which 1 have generally translated by ^* corresponding.'^ But here it is followed by a dative. 5;4£X(^a r«r fiXeif *' Acmoio^Mi viih the wholes ," instead of being used ab^iotutely. The meaning can therefore here be nothing else but 'Mn the same ratio with" or "proportional to the wholes"^ and Euclid seems to recognise that he is making a special use of the word, because he eitplains it lower down (1. 46) : "the triangles are homolc^ous to the wholes, thai is, in such manner that the triangles are proportional, and AB£j EBC, ECD are ante- cedents, while FGL, LGH, I. UK are their consequents," 49. iv6fu¥a rr0Twf, " iA^tr consequents," is a little awkward, but may be supposed to indicate which triangles curtespond to which ai consctjuent to antecedent. . ,., , . An alternative proof of the second part of this proposition given after the Porisms is relegatetj by August and Heiberg to an Appendix as an interpolation. It is shorter than the proof in the text, anti is the only one given by many editors, including Clavius, Billingsley, Barrow and Simson. It runs as follows: " We will now also prove that the triangles are homolc^ous in another and an easier manner. Again, let the polygons ABCDE, FGHKL be set out, and let BE, EC, GL, Z//^ be joined I say that, as the triangle A BE is to FGL, so is EBC to LGH and ODE to HKL. For, since the triangle ABE is similar to the triangle FGL, the triangle ABE has to. the triangle FGL a ratio duplicate of that which BE has to GL, VI, 30, ai] PROPOSITIONS ao, it *$g. For the same reason also the trkngte BEC has to the triangle GLIf a ratio duplicate of that i^ich £E has to GL. Therefore, as the triangle ABE is to the triangle EG£, so is BEC 10 GZff. Again, since the triangle EBC \s similar to the triangle LGH, EBC has to LGH a ratio duplicate of that which the straight line CE has to HL. For the same reason aiso the triangle ECD has to the triangle LHK a ratio duplicate of that which CE has to HL. Therefore, as the triangle EBC is to LGH, so is ECD to LHK. But it was proved that, as EBC is to LGH, so also is ABE to FGL. Therefore also, as ABE is to FGL, so is BEC to GLH and ECD to LHK. • . •• ■ ' It Q. E. D. Now Euclid cannot fail to have noticed that the second part of his proposition could be proved in this way. It seems therefore that, in giving the other and longer method, he deliberately wished to avoid using the result of VI. 19, preferring to prove the first two parts of the theorem, as they can be proved, independently o( any relation between the areas of similar triangles. The first part of the Porism, stating that the theorem is true oi quadriiaitrals, would be superfluous but for the fact that technically, according to Book i. Def, 19, the term "polygon "{or figure of many sides, iroXuTrXtupoi') used in the enunciation of the proposition is confined to rectilineal figures of more than four sides, so that a quadrilateral might seem to be excluded. The mention of the triangle in addition fills up the tale of " similar rectilineal figures." The second Porism, Theon's interpolation, given in the text by the editors, but bracketed by Heiberg, is as follows : "And, if we talce O a third proportional to AB, EG, then BA has to O a rtUio duplicate of that which AB has to EC. But the polygon has also to the polygon, or the quadrilateral to the quadrilateral, a ratio duplicate of that which the corresponding side has to the corresponding side, that is AB to EG; and this was proved in the case of triangles aiso ; so that it is also manifest generally that, if three straight lines be proportional, as the first is to the third, so will the figure described on the first be to the similar and similarly described figure on the second." Proposition 21. • . ■ • Figures which are similar to the same rectilineal figure are also similar to one another. For let each of the rectilineal figures A,B]x. similar to C\ 1 say that A is also similar to ^. «4o BOOK VI [vi. 31, It For, since A is similar to C * it is equiangular with it and has the sides about the equal angles proportional. [vi. Def. i] tn Again, since B is similar to C, it is equiangular with it and has the sides about the equal angles proportional, Therefore each of the figures A, B is equiangular with C and with C has the sides about the equal angles proportional; therefore A is similar to B, Q. E, D, It will be observed that the text above omits a. step which the editions generally have before the final inference " Therefore A is similar to B." The words omitted are "so that A is also equiangular with B and [with B] has the sides about the equal angles proportional." Heiberg follows P in leaving them out, conjecturing that they may be an addition of Th eon's. Proposition 22. J/ /our straight lines be proportional, the rectilineal figures similar and similarly described upon them -will also be pro- portional ; and, if the rectilineal figures similar and similarly described upon them be proportional, the straight lines will themselves also be proportional. Let the four straight lines AB, CD, EF, GH be pro- portional, so that, as ^.ff is to CD, so is EFxo GH, and let there be described on AB, CD the similar and similarly situated rectilineal figures KAB, LCD, and on EF, GH the similar and similarly situated rectilineal figures J/A, NH; I say that, as KAB is to LCD, so is MF to NH. For let there be taken a third proportional O to AB, CD, and a third proportional fi to EF, GH. [vi. n] VI. »a] PROPOSITIONS ai, »a J4|, Then since, as AB is to CD, so is EF to GH, and, as CZ? is to O, so is GH to /*, therefore. Mr aequali, as Wi? is to O, so is EF to P. [v. u] But, as -f4^ is to t>, so is AVJ^ to ZCZ>, and, as EF Is to /", so is MF to A^^; ^'''' "'' **''^ therefore also, as KAB is to Z,CA so is MF to A''//, [v. n] /7 Next. let MF be to A^//^ as KAB is to ZLCZ? ; I say also that, as ^4 5 is to CD, so is EFto GH. For, if EF is not to GH as ^^ to CZ?, let j£"/^be to jjj? as AB to CA [vi. u] and on QR let the rectilineal figure SR be described similar and similarly situated to either of the two MF, NH. [vi. i8] Since then, as AB is to CD, so is EF to QR, and there have been described on AB, CD the similar and similarly situated figures KAB, LCD, '- and on EF, QR the similar and similarly situated figures MF, SR, therefore, as KAB is to LCD, so is MF to SR. But also, by hypothesis, as KAB is to L CD, so is MF to NH ; ' ■ therefore also, as MF is to SR, so is MF to A''//^ [v. 1 1] Therefore MF has the same ratio to each of the figures NH,SR; therefore NH is equal to SR. [v. 9] But it is also similar and similarly situated to it ; therefore GH is equal to QR. S4» BOOK VI '< ' [vi. ** And, since, as AB is to CD, so is £F to QJ?, while QH is equal to G//, therefore, as AB is to CD, so is EFto GH. " '" Therefore etc. A , ' • / t . Q. E. D. The second assumption in the first step of the first part of the proof, vi^. that, as CD is to O, so GM to J", should perhaps be explained. It is a deduction [by v. ii] from the facts that -. AMis to CJ? as CD to O, £I!'is to Gffas GUto P, Mid •- AB is to CD as EFta GH. The defect in the proof of this proposition is well known, namely the assumption, without proof, that, because the figures NH, SR are equal, besides being similar and similarly situated, their corresponding sides GH, QR are equai. Hence the minimum addition necessary to make the proof complete is a proof of a lemma to the effect that, iffim similar figurn are also equal, any pair of corresponding sides are equal. To supply this lemma is one alternative ; another is to prove, as a preliminary proposition, a much more general theorem, viz. that, tf the duplicate ratios of two ratios an equal, the tnio ratios are themselves equal. When this is proved, the second part of vi, 12 is an immediate infeience from it, and the effect is, of course, to substitute a new proof instead of supplementing Euclid's. I. It is to be noticed that the lemma requitt:d as a mini mum is very like what is needed to supplement vi. 28 and 19, in the proofs of which Euclid assumes that, if tivo similar parallelograms are unequal, any side in the greater is greater than the corresponding side in the smaller. Therefore, on the whole, it seems preferable to adopt the alternative of proving the simpler lemma which will serve to supplement all three proofs, vii, that, if of two similar rectilineal figures the first is greater than, equal to, or less thaft, the second, any side of the first is greater than, equal to, or less than, the corresponding side of the second respectively. The case of equality of the figures is the case required for vi. t2 ; and the proof of it is given in the Greek text after the proposition, Since to give such a " lemma " after the proposition in which it is required is contrary to Euclid's manner, Heiberg concludes that it is an interpolation, though it is earlier than TheotL The lemma runs thus ; "But that, if rectilineal figures be equal and similar, their corresponding sides are equal to one another we will prove thus. Let Nff, SR be equal and similar rectilineal figures, and suppose that, as HG is to GN, so is RQ to QS', I say that RQ is equal to HG. For, if they are unequal, one of them is greater; let ^Q be greater than HG. VI. sa] PROPOSITION 31 _s4S Then, since, as ^0 is to QS, so is I/G to GNi alternately also, as ^Q is to //G, so is QS to GJVi and Q£ is greater than J/G ; therefore QS is also greater than GN; so that /IS is also greater than HN*. But it is also equal : which is impossible; ' " ' Therefore QR is not unequal to GH\ •. vk D (•' therefore it is equal to it," ^«^f.^ ,> [The step marked • is easy to see if it is remembered that it is only necessary to prove its truth in the case of triangles (since similar polygons are divisible into the same number of similar and similarly situated triangles having the same ratio to each other respectively as the polygons have). If the triangles be applied to each other so that the two corresponding sides of each, which are used in the question, and the angles included by them coincide, the truth of the inference is obvious.] The lemma might also be arrived at by proving that, ^« ratio is greater than a ratio of equality, tht ratio which «> its dupHeate is also greater than a ratio of equality ; and if ike ratio lohich is duplicate of anotfur ratio is greater than a ratio of equally, the ratio of which it is the duplicate is also greater than a ratio of equality. It is not difficult to prove this from the particular case of v. 25 in which the second magnitude is equal to the third, i.e. from the fact that in this case the sum of the extreme terms is greater than double the middle term. II. We now come to the alternative which substitutes a new proof for the second part of the proposition, making the whole proposition an immediate inference from one to which it is practically equivalent, viz. that (i) // duo ratios be equal, their duplicate ratios are equal, and (2) cott' versely, if the duplicate ratios of two ratios be equal, the ratios are equal The proof of part (i) is after the manner of Euclid's own proof of the first part of VI, a*. Let j1 be to .5 as C to A and let Jf be a third proportional to A, B, and Fa third proportional to C, D, so that A '\% \!0 B »& B to X, and C is to Z) as 2J to Y\ whence A'\%\a X in the duplicate ratio of A to B, and C is to Y in the duplicate ratio of C to D. Since j4 13 to i? as C is to A and ^ is to Jf as /i is to .5, i.e. as C is to Z*, , , IV. Ill i.e as /J is to K, . ^ ^ therefore, ex aequali, /f is to Jf as C is to K -• Part (2) is much more difficult and is the crux of the whole thing. Most of the proofs' depend on the assumption that, B being any magnitude and P and Q two magnitudes of the same kind, there does exist a magnitude A which is to S in the same ratio as /' to Q, It is this same assumption «44 '■'■ BOOK VI [vi. ai which makes Euclid's proof of v. i8 illegitimate, since it is nowhere proved in Book v. Hence any proof of the proptosition now in question which involves this assumption even in the case where B, P, Q are all straight lines should not properly be given as an addition to Book v. ; it should at least be postponed until we have leamt, by means of vi. iz, giving the actual construction of a fourth proportional, that such a fourth proportional exists. Two proofs which are given of the proposition depend upon the following lemma. ^A, B, C ^ tAru magnitudes of ofte kind, and D, E, F three Mitgnitudes of one kind, then, if Ike ratio of A fa B is greater than that of D toE, i ' and tht ratio of "^ to Q, greater than that of 'R to'F, ' ex aequali, the ratio of A to C is greater than thai of V to F. One proof of this does not depend upon (he assumption referred to, and therefore, if this proof is used, the theorem can be added to Book v. The proof is that of Hauber (Camerer's Euclid, p. -358 of Vol. u.) and is reproduced by Mr H. M. Taylor. For brevity we will use symbols. Take equimultiples m^, mD of A, D and nB, nE of B, E such that mA>nB, but mD'^nE. Also let pB, pE be equimultiples of B. E and qC, qP equimultiples of C, .f such that pB>qC, hai pElf-gF. Therefore:, multiplying the first tine by^ and the second by n, we have pmA>pnB,pmD1(-pnE, i. and npS>nqC,npE1f-nqF, whence pmA>nqC, pmDI^nqF. ' \ Now P*nA, pmD are equimultiples of mA, mU, and nqC, ffl?/^ equimultiples of qC, qF. Therefore [v. 3] they are respectively equimultiples of A, D and of C, F. Hence [v. Def. 7] A : C>D : F. Another proof given by Claviua, though depending on the assumption referred to, is neat Take G such that GiC = E'.F. • ■- A B A Therefore and Therefore «i . ■.C>G.C, B>G. ■.G>AiB. [v. 13] [v. 10] [v. 8] VI. ai] PROPOSITION 22 145 But A ; 3>DiE. Therefore, a fortiori. A : G>D:E. Suppose ^ taken such that H: G= D:B. Therefore A>H. Hence A : : C>H'. C. But H. .G = D:£, G . C^E:F. Therefore, ex aequali^ J5f ; C=£}.K Hence A : \C>D.R [V. '3. .0] [V .8) [V. 2.] [V- ■3] Now we can prove that Ratios of whUk equai ratios are duplitate are equal. Suppose that A : .B = B: C, and D :£ = £ :iS and further that A : ; €=£>: E it is required '0 prove that A B^D-.E. For, if not, one of the ratios must be greater than the othur. I>et A : B\x^t greater. Then, since A : B = B : C^ and £>:E = E:E, while A: B>D: E, it follows that B: C>E\F. [v- 13] Hence, by the lemma, ex aequali, A : C>D -.F, which contradicts the hypothesis. Thus the ratios A : B and D : E cannot be unequal j that is, they are equal. Another proof, given by Dr Lachlan, also assumes the existence of a fourth proportional, but dei:)ends upon a simpler lemma to the effect that // is impussible that two different ratios can have the same duplicate ratio. For, if possible, let the ratio ^ : ^ be duplicate both ol A -.X and A : V, so that [V.8] [v. II, 13] [V. lo] A ■.X = X ■B, and A : y= : V ■B. Let X be greater than y. Then A: x< A : y-. that is, X . B< Y: -e, or x< Y. But X is greater than Y: which IS a Lbsurd, etc. Hence x= y. %t6 BOOK VI Now suppose that A : B^B: C, JDiE = E .F, and A : C = D:F. To prove that A\B = D:E. If this is not so, suppose that A:B = D:Z. Since A.C^D.F, therefore, inversely, C:A=F:D. Therefore, «c aequali. C: JS = F:2, or, inversely. B: C = Z: F. Therefore A.B = Z:F. But A : B ^ D : Zy^yj hypothesis. Therefore D:Z=Z. F. Also, by hypothesis, D:E=E:F; whence, by the lemma, E = Z. Therefore A-.B-^D-.E. [VI. [v. J a] [v. m] [V. ..J De Morgan remarks that the best way of remedying the defect in Euclid is to insert the proposition {the lemma to the last proof) that // is imposiihU thai two differml ratios can have the same duflitate ratio, "which," he says, "immediately proves the second (or defective) case of the theorem." But this seems to be either too much or too little : too much, if we choose to make Che minimum addition to Euclid (for that addition is a lemma which shall prove that, if a duplicate ratio is a ratio of equality, the ratio of which it is duplicate is also one of equality), and too little if the proof is to be altered in the more fundamental manner explained above, I think that, if Euclid's attention had been drawn to the defect in his proof of VI. 2 2 and he had been asked to remedy it, he would have done so by supplying what I have called the minimum lemma and not by making the more fundamental alteration. This I infer from Prop, 24 of the Data, where he gives a theorem corresponding to the proposition that ratios of which equal ratios, are duplicate are equal. The proposition in the Data is enunciated thus : If three straight tines be proportional, and the first have to the third a given ratio, it wilt also have to the second a given ratio. A, B, C being the three straight lines, so that A .B = B.C, and A : C being a given ratio, it is required to prove that A : B vi also a given ratio. Euclid takes any straight line D, and first finds another, F, such that D .F=A -.C, whence D : F must be a given ratio, and, as ^ is given, F is therefore given. Then he takes E a mean proportional between D, P, so that D .£ = E:F. VI. J J, 23] PROPOSITIONS 2t, 13 f||> It follows [vi. 17] that the rectangle D, Ph equal to the square on E. But D, Fzxe both given ; therefore the square on E is given, so that E is also given. [Observe that De Morgan's lemma is here assumed without proof. It may be proved {ij as it is by I>e Morgan, whose proof is that given above, p. 345, (i) in the manner of the "minimum Semma," pp. 342 — 3 above:, or (3) as it is by Proclus on i. 46 (see note on that proposition).] Hence the ratio D \ E\s given. Now, since A : C= D : J<, and A : C= (square an A): (rect. A, C), while D : F~ (square on D') : (tect. D, F), [vi. i] therefore (square on A) ; (rect. A, C)~ {square on D) • (rect. D, F). [v. 11] But, since A -. B = B : C, (rect. A, C) - (sq. an S); [vi. 17] and (rect. £>, F) = (sq. on E}, from above ; therefore (square on A) : (square on ^) = (sq, on i?) : {sq. on £). ' ■ ' Therefore^ says Euclid, • 1 w A;B = D:E, that is, ht assumes the truth of •i\. a fat squarts. Thus he deduces his proposition from vi, 21, instead of proving vi. 33 by means of it (or the corresponding proposition used by Mr Taylor and Or I^chlan). ■i/u . .' .• > .:,.•-. Proposition 23. '■' ' "' Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides. Let AC, CF be equiangular parallelograms having the angle BCD equal to the angle ECG ; jl say that the parallelogram AC has to the parallelogram CF the ratio (;ompounded of the ratios of the sides. 7 K B' U M For let them be placed so that BC is in a straight line with CG ; therefore DC is also in a straight line with CE. I Let the parallelngram DG be completed ; let a straight line K be set out, and let it be contrived that, as BC is to CG, so is i^ to Z. , and, as DC is to CE, so is L to M. [vi. 12] 348 BOOK VI [vi. »3 Then the ratios of K to L and oi L, Xo M are the same ij as the ratios of the sides, name]y of BC to CG and of DC to CE. But the ratio of A' to ^ is compounded of the ratio of K to L an<i of that of Z to M; so that JC has also to M the ratio compounded of the ratios ao of the sides. Now since, as BC is to CG, so is the parallelogram AC to the parallelogram C//, [vi. i] while, as BC is to CG, so is A" to Z, therefore also, as K is to L, so is AC to C//. [v. n] as Again, since, as Z?C is to C£, so is the parallelogram C// to CjS [vi. r] while, as DC is to C£, so is L to ^, therefore also, as Z. is to M, so is the parallelogram C/f to the parallelogram CK [v, 1 1] 30 Since then it was proved that, as /f is to Z, so is the parallelogram AC to the parallelogram C//, and, as Z is to ^, so is the parallelogram CH to the parallelogram CB", therefore, mt ae^uali, as A' is to M, so is AC to the paralle lo- ss gram CF. But A" has to M the ratio compounded of the ratios of the sides ; therefore A C also has to CF the ratio compounded of the ratios of the sides. 40 Therefore etc, Q. E. D. 1,6, 19, 36. the ratio compounded of the ratios of the sides, M^w riy rirrmifunr 4t T«>' TrXtvpi^ which, nieining literally " the ratio coropounded 0/ (it lijes," is negligently writlen here and commonly for Xiryav rir rvyKtif^trov in t«v tup i[\xvpi4» (sc. X^yuv}* II. let it be contrived that, as BC is to CO, so is K to L, The Greek phraiie in of the usual ter^e kind, untranslatable literally : xol yeyofirw ajt ftir ^ EF Tpit rip TE, offrwi 17 K irpij rf A, the words meaning "and let (there) be made, u BC to CG, sa Kio Z," where L is the straight line which has to be constructed. The second tJefinirion of the Daia says that A ratio is said to fe given if we can find (Tropi'o-ao-flot) [another ratio that />] the same with it. Accordingly VI, 13 not only proves that equiangular parallelograms have to one another a ratio which is compounded of two others, but shows that that ratio is "given" when its component ratios are given, or that it can be represented as a simple ratio between straight lines. VI. i3] PROPOSITION »3 94^ Just as vt. 13 exhibits the o[>eration necessary for eomfounding two ifttios, a proposition (8) of the Data indicates the operation by which we may divide one ratio hy another. The proposition proves that Thingt which have a given raiio to the same thing have also a grven ratio to one another. Euclid's procedure is of course to comf>ound one ratio vrith the inverse of the other ; but, when this is once done and the result of Prop. 8 obtained, he uses the result in the later propositions as a substitute for the method of composition. Thus he uses the division of i^tios, instead of composition, in the propositions of the Data which deal with the same subject-matter as VL 13. The effect is to represent the ratio of two equiangular parallelc^rams as a ratio between straight lines one of which is one side of one of the paraiMograms, Prop. 56 of the Data shows us that, if we want to express the ratio of the parallelogram AC Xx> the paraDelogram CF in the figure /■ ov. ur.' '.I ( ..I '■■ ' .... ■■ , A'' of VI. 23 in the form of a ratio in which, for exampie, the side flC is the antecedent term, the required i-atio of the parallelograms is BC ; X-, where DC:CE=CG: X, or A" is a fourth proportional to DCarA the two sides of the parallelogram CF. Measure CK along CB, produced if necessary, so that DC .CE=CG . CK (whence CK is equal to X). * [This may be simply done by joining DG and then drawing EK parallel to it meeting CB in /T.] Complete the parallelogram AK. Then, since DC : CE ^ CG : CK, the parallelc^rams DK, C^are equal. [vi. 14] Therefore (AC).{CF) = {AC).(DK) > ■ [v. 7] = BC .CK ' [VI. i] ^BC.X. Prop. 68 of the Data uses the same construction to prove that. If two equiangular parallelograms havt to one another a given ratio, and one side have to one side a given ratio, the remaining side will also have to the remaining side a given ratio. I do not use the figure of the Data but, for convenience' sake, I adhere to the figure given above. Suppose that the ratio of the parallelograms is given, and also that of CD to CE. Apply to CD the parallel<^ram DK equal to CE and such that CK, CB coincide in direction. [i. 45] TTien the ratio of AC to KD is given, being equal to that of ACio CF. And {AC) : (KB) = CB : CK; therefore the ratio of CB to CK is given. >SO BOOK VI [vi. ^3 But. since KD= CF, Cn-.CE^CG: CK. [vi. 14] Hence CG : CK'xs a given ratio. And CB : CK was proved to be a given ratio. Therefore the ratio of CB to CG is given. \Data, Prop. 8] lastly we may refer to Prop. 70 of the Data, thL- first part of which proves what corresponds exactly to vi. 33, namely that, // in ttvo fquiangular paral- lelograms Ihe sides containing tht equal angles ham a given ratio to one atmlher [i.e. one side in one to one side in the other], the paralUlograms themselves will also have agiivn rat is to one another. [Here the ratios of BC to CG and of CD to CE are given.] The construction ts the same as iti the last case, and we have KD equal to CF^ so that CD . CE=CG : CK. [vi. 14] But the ratio of CD to CE is given ; therefore the ratio of CG to C^is given. And, by hypothesis, the ratio of CG to CB is given. Tlierefore, by dividing the ratios [ZJd/ir, Prop. 8], we see that the ratio of CB to CK^ and therefore [vj, i] the ratio of .^C to DK, or of AC to CF, is given. Euchd extends these propositions to the case of two parallelograms which have givfn but not equal angles. Pappus (v[i. p. 928) exhibits the result of vi. 13 in a different way, which throws new light on compounded ratios. He proves, namely, that a paratUlograin is to an equiangnlar parallelogram as the rectangle amtained hy tlu adjacent sides of the first is to the rectangle contained by the adjacent sides oj the second, A o C E t-et A C, OF he equiangular parallelograms 01 r the bases BC, EF, and let the angles at .5, £ he equal. Draw perjjendiculars AG, DH \.o BC, £/" respectively. Since the angles at B, G are equal to those at E, H, the triangles ABG, DEHsliu equiangular. ' Therefore BA : AG ED: DM. [vi. 4] But BA 'AG = (rect. BA, BC) : (rect. AG, BC), and ED : DH ^ (rect, ED, EF) : (rect DH, EF). [vi, i] Therefore [v. [ 1 and v. 16] (TticX.. AB, BC) : (rect. DE, EF) = (rect. AG, BC) : (rect DH, EF) = {AC).{DF). Thus it is proved that the ratio compounded of the ratios A3 : DE and BC : EF is equal to the ratio of the rectangle AB, BC to the rectangle DE, EF. VL 83, 34] PROPOSITIONS 23, 14 *St Since each parallelogram in the figure of the proposition can be divided into pairs of equal triangles, and all the triangles which are the halves of either parallelogram have two sides respectively equal and the angles included by them equal or supplementary, it can be at once deduced from vt. 23 (or it can be independently proved by the same method) that triangles which have ont angle of the one equal or supplementary to one angle of the other are in (he ratio (ompoa tided of the ratios of the sides about the eqval or supplementary angles. Cf- Pappus vii. pp. 894—6. VI. 23 also shows that rectangles, and therefore parallelograms or triangles, are to one another in the ratio compounded of the ratios of theit bases and heights. The converse of vi. 23 is also true, as is easily proved by reductio ad absurdum. More generally, if two parallelograms or triangles are in the ratio compounded of the ratios of two adjacent sides^ the angles included by those sides are either equal or supplementary. Proposition 24. In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another. Let ABCD be a parallelogram, and AC its diameter, and let EG, HK be parallelograms about AC\ I say that each of the parallelograms EG, HK is similar both to the whole ABCD and to the other. For, since EF has been drawn parallel to BC, one of the sides of the triangle ABC, proportionally, as BE is to EA, so is CF to FA. [vi. 2] Again, since FG has been drawn parallel to CD, one of the sides of the triangle ACD, proportionally, as CF\^ to FA, so is DG to GA. [vi. 2], But it was proved that, as CF is to FA, so also is BE to EA ; therefore al.so, as BE is to EA, so is DG to GA, and therefore, componendo, as BA is to AE, so is DA to AG, \y. 18] and, alternately, as BA is to AD, so is EA to AG. [v. 16] Therefore in the parallelograms ABCD, EG, the sides about the common angle BAD are proportional. And, since GF is parallel to DC, »j^ , BOOK VI [vi. 24 the angle AFG is equal to the angle DC A ; and the angle DAC is common to the two triangles ADC, AGF, therefore the triangle ADC is equiangular with the triangle AGF. For the same reason the triangle ACB is also equiangular with the triangle AFE. and the whole parallelogram ABCD is equiangular with the parallelogram EG. Therefore, proportionally, as AD is to DC, so is v4G^ to GF, as DC is to CA, so is GF to FA., as ^C is to CB, so is AF to FE, and further, as CB is to BA, so is FE to EA. And, since it was proved that, as DC is to CA, so is GF to /v?, and, as AC is to C^, so is AF to /^£, therefore, ex aequali, as Z?C is to CB, so is GF to 7^/5'. [v. j^J Therefore in the parallelograms ABCD, EG the sides about the equal angles are proportional ; therefore the parallelogram ABCD is similar to the parallelo- gram EG. [vi. Def. i] For the same reason the parallelogram ABCD is also similar to the parallelogram KH', therefore each of the paralleJograms EG, HK is simitar to ABCD. But figures simlliar to the same rectilineal figure are also similar to one another ; [vi. ai] therefore the parallelogram EG is also similar to the parallelo- gram HK. Therefore etc. Q, E, D. Simson was of opinion that this proof was made up by some unskilful editor out of two others, the first of which proved by parallels (vi. 2) that the sides about the common angle in the parallelograms are proportional, while the other used the similarity of triangles {vi. 4.). It is of course true VI. 24. 25] PROPOSITIONS 34, 35 »S3 that, when we have proved by vi. 2 the fact that the sides about the common angle are proportional, we can infer the proportionality of the other sides directly from 1, 34 combined with v. 7. But it does not seem to me unnatural that Euclid should (i) deliberately refrain from making any use of t. 34 and (3) determine beforehand that he would prove the sides proportional in a difinite ordtr beginning with the sides EA, AG and BA, AD about the common angle and then taking the remaining sides in the order indicated by the order of the letters A, G, F, E. Given that Euclid started the proof with such a fixed intention in his mind, the course taken presents no difficulty, nor is the proof unsystematic or unduly drawn out. And its genuineness seems to me supported by the fact that the proof, when once the first two sides about the common angle have been disposed of, follows closely the order and method of vi. 18. Moreover, it could readily be adapted to the more general case of two p>olygons having a common angle and the other corresponding sides respectively parallel. The parallelograms in the proposition are of course similarly situated as well as similar; and those "about the diameter" may be "about" the diameter produced as well as about the diameter itself From the first part of the proof it follows that parallelograms which have one an^te equal to one angle and the sides about those angles proportional are similar. Prop. 26 is the converse of Prop. 24, and there seems to be no reason why they should be separated as they are in the text by the interposition of VI. %i. Campanus has vi. 24 and 26 as vi. 22 and 23 respectively, vi. 23 as VI, 14, and VI. 35 as we have it. Proposition 25. To construct one and the same figure similar to a given rectilineal figure and equal to another given rectilineal figure. Let ABC be the given rectilineal figure to which the figure to be constructed must be similar, and D that to which it must be equal ; thus it is required to construct one and the same figure similar to ABC and equal to D. Let there be applied to BC the parallelogram BE equal to the triangle ABC [1. 44], and to CE the parallelogram CM equal to D in the angle FCE which is equal to the angle CBL. [1. 45] »$♦ BOOK Vr [vi. 35 • Therefore BC is in a straight line with CF, and LE with EM. Now let GH be taken a mean proportional to BC, CF [vL 13J, and on GHlel KGHh& described similar and similarly situated to ABC. \y\. 18] Then, since, as BC is to GH, so is GH to CF, and, if three straight lines be proportional, as the first is to the third, so is the figure on the first to the similar and similarly situated figure described on the second, [vi. 19, For,] therefore, as BC is to CF, so is the triangle ABC to the triangle KGH. But, as BC is to CF, so also is the parallelogram BE to the parallelogram EF. [vi. i] Therefore also, as the triangle ABC is to the triangle KGH, so is the parallelogram BE to the parallelogram EF \ therefore, alternately, as the triangle ABC is to the parallelo- gram BE, so is the triangle KGH to the parallelogram EF. [v. 16] But the triangle ABC is equal to the parallelogram BE ; therefore the triangle jf 6'//' is also equal to the parallelogram EF. But the parallelogram EF is equal to D \ therefore KGH is also equal to D, And KGH is also similar to ABC. Therefore one and the same figure KGH has been con- structed similar to the given rectilineal figure ABC and equal to the other given figure D. Q, E, D. 3. to which the flgiire to be const ructed muBt be similar, literally " lo which it Is requiied to construct (one) Eimiisr," ^ itiinoaio (rwrriiffMSm. This is the highly important problem which Pythagoras is credited with having solved. Compare the passage from Plutarch {Symp. vtu. 2, 4) quoted ■n the note on i. 44 above, Vol, 1. pp. 343 — 4. We are bidden to construct a rectilineal figure which shall have the form of one and the siu of another rectilineal figure. The corresponding proposition of the Data, Prop. 55, asserts that, "if an area {xtapiav) be given in form {(iSii) and in magnitude, its sides will also be given in magnitude." Simson sees signs of corruption in the text of this proposition also. In the first plac^ the proof speaks of the triangle ABC, though, according to the enunciation, the figtire for which ABC is taken may be any rectilineal figure, tutfuypafifiop "rectilineal figure" would be more correct, or <ISo9, "figure"; the mistsjie, however, of using rpiyavoy is not one of great importance, being no VI. »5. »6] PROPOSITIONS 25, a6 »S5 doubt due to the accident by which the tigure was drawn as a. triangle in the diagram. The other observation is more important. After Euclid has proved that (fig. ABC) ; (fig. JCG/T) = {B£) : {Ef), he might have inferred directly from v. 14 that, since ABC is equal to BE, KGH is e<jual to EF. For v. 14 includes the proof of the fact that, if A is to .^ as C IS to D, and A is etjual to C, then B is equal to D, or that of four proportional magnitudes, if the first is equal to the third, the second is equal to the fourth. Instead of proceeding in this way, Euclid first permutes the proportion by v. 16 into (fig. ABC) : {BE) = (fig. KGH) : (EF), arid then infers, as if the inference were easier in this form, that, since the Jlftt is equal to the secend, the third is equal to the fourth. Yet there is no proposition to this effect in Euclid. The same unnecessary step of permutation is also found in the Greek text of xi. 23 and xii. 2, 5, ir, 12 and 18. In reproducing the proofs we may simply leave out the steps and refer to v. 14. Proposition 26. If frotn a parallelogram there be taken away a parallelo- gram similar and mnilarly situated to the whole and having a common angle with it, it is about the same diameter with the whole For from the parallelogram A BCD let there be taken away the parallelogram AF similar and similarly situated to ABCD, and having the angle DAB common with it ; I say that ABCD is about the same diameter with AF. For suppose it is not, but, if possible, let AHCh& the diameter < oiABCD > , let GF be produced and carried through to ff, and let /fK be drawn through // parallel to either of the straight lines AD, BC. [i. 31] Since, then, ABCD is about the same diameter with KG, therefore, as DA is to AB, so is GA to AK. [vi. 24] But also, because of the similarity of ABCD, EG, as DA is to AB, so is GA to AE ; therefore also, as GA is to AK, so is GA to AE. [v. 1 1] Therefore GA has the same ratio to each of the straight lines AK, AE. •S« •> BOOK VI [VI. !t6 Therefore A£ is equal to ^A' [v. 9], the less to the greater: which is impossible. Therefore A BCD cannot but be about the same diameter with AF\ therefore the parallelogram A BCD is about the same diameter with the parallelogram AF. Therefore etc. Q. E. D. " For suppose it is not, but, if possible, let AHC be the diameter." WKat is meant is " For, if AFC is not the diameter of the paralleiogram AC, let AlfC be its diameter." The Greelt text has fo-Tut amav Sta/wrpiK ij AOT; but clearly avrwv is wrong, as we cannot assume that one straight line is the diameter of both parallelograms, which is just what we have to prove. F and V omit the auTiui-, and Heiberg prefers this correction to substituting aurou after Peyrard. I have inserted " < of A BCD > " to make the meaning clear. If the straight line AHC does not pass through F, it must meet either GF (X £?/' produced in some point H. The reading in the text "and let GF be produced and carried through io H " (uat im^X-ifivxta, ^ HZ &7xSm Jjrt TO ®) corresponds to the supposition that H is on GF produced. The words were left out by Theon, evidently because in the figure of the mss. the letters E, Z and K, were interchanged, Heiberg therefore, following August, has preferred to reKiin the words and to correct the figure, as well as the passage in the text where AE, AK were interchanged to be in accord with the MS. figure. It is of course possible to prove the proposition directly, as is done by Dr Lachlan. Let AF, ACh^ the diagonals, and let us matte no assumption as to how they fall. Then, since EF\s parallel to AG and therefore to BC, the angles AEF, ABC are equal. c And, since the paraRetograms are similar, AE : EF=AB : BC. [vi. Def. i] Hence the triangles AEF, ABC are similar, [vi. 6] and therefore the angle FAE is equal to the angle CAB. Therefore AFfalh on AC. The proposition is equally true if the parallelogram which is similar and similarly situated to the given parallelogram is not " taken away" from it, but is so placed that it is entirely outside the other, while two sides form an angle vertically opposite to an angle of the other. In this case the diameters are not "the same," in the words of the enimciation, but are in a straight line with one another. This extension of the proposition is, as will be seen, necessary for obtaining, according to the method adopted by Euclid in his solu- tion of the problem in vi. s8, the second solution of that problem. vj. a?} PROPOSITIONS 26, 27 Proposition 27. 0/ all ike parallelograms applied to the same straight line and deficient by parallelogrammic figures similar and similarly situated to that described on the half of the straight line, that parallelogram is greatest which is applied to the half of the straight line and is similar to the defect. Let AB be a straight line and let it be bisected at C; let there be applied to the straight line AB the parallelogram AD deficient by the parallelogrammic figure DB described on the half of AB, that is, CB; I say that, of all the parallelograms applied to AB and deficient by parallelogrammic figures similar and similarly situated to T)B, A J? is greatest. For let there be applied to the straight line AB the parallelogram AF deficient by the parallelogrammic figure FB similar and similarly situated to DB; I say that AD is greater than AF. P'or, since the parallelogram DB is similar to the parallelo- gram FB, they are about the same diameter. [vi. 36] Let their diameter DB be drawn, and let the figure be described. Then, since CF is equal to F£, _ [i. 43] and FB is common, therefore the whole C// is equal to the whole /CF. But C// is equal to CG, since AC is also equal to CB, [I. 36] Therefore GC is also equal to F/l. Let CF be added to each ; therefore the whole AFis equal to the gnomon LMN\ so that the parallelogram DB, that is, AD, is greater than the parallelogram AF. Therefore etc. as* BOOK VI [vi. aj We have already (note on i. 44) seen the significance, in Greek geometry, of the theory of " the application of areas, their exceeding and their falling- short." In I. 44 it was a question of " applying to a given straight line (exactly, without 'excess' of 'defect') a parallelogram equal to a given rectilineal figure, in a given angle," Here, in vi. 27 — 29, it is a question of parallelograms applied to a straight line but "deficient (or exceeding) by paralUiograms similar aiid similarly situated to a given parallelogram." \^ Apart from size, it is easy to construct ^ 7 any number of parallelograms " de- r / '^^ 7" ficient" or "exceeding" in the manner / / / ^\A l described. Given the straight line /' e F ^(™...... AB to which the parallelogram has to L . ('....^^s/ be applied, we describe on the base ^ ^ '^\ CB, where C is on AB, or on BA produced beyond A, any parallelogram " similarly situated " and either equal or similar to the given parallelogram (Euclid takes the similar and similarly situated parallelogram on half the line), draw the diagonal BD, take on it (produced if necessary) an> points as E, K, draw EF, or KL, parallel to CI) to ineet AB ot AB produced and complete the parallelograms, as AH, ML, If the point E is taken on BD or BD produced beyond Z>, it must be so taken that EF meets AB between A and B. Otherwise the parallelogram AE would not be applied to AB itself, as it is required to be. The parallelograms BD, BE, being about the same diameter, are similar [vi, 24], and BE is the defect of the parallelogram AE relatively to AB, AE is then a parallelogram applied ta AB but deficient by a parallelogram similar and similarly situated to BD, If A* is on D3 produced, the parallelogram BK is similar to BD, but it is the excess of the parallelogram AK relatively to the base AB. AK h a. parallelc^ram applied to AB but exceeding by a parallelogram similar and similarly situated to BD. Thus it is seen that BD produced both ways is the locus of points, such 9S E OT K, which determine, with the direction of CD, the position of A, and the direction of AB, parallelograms applied to AB and deficient or exceeding by parallelograms similar and similarly situated to the given parallelogram. The importance of vi. 27 — 29 from a historical point of view cannot be overrated. They give the geometrical equivalent of the algebraical solution of the most general form of quadratic equation when that equation has a real and positive root. It will also enable us to find a real negative root of a quadratic equation ; for such an equation can, by altering the sign of x, be turned into another with a real positive root, when the geometrical method again becomes applicable. It will also, as we shall see, enable us to represent both roots when both are real and positive, and therefore to represent both roots when both are real but either positive or negative. The method of these propositions was constantly used by the Greek geometers in the solution of problems, and they constitute the foundation of Book X. of the Elements and of ApoUonius' treatment of the conic sections. Simsons observation on the subject is entirely justified. He says namely on VI. 28, 29: "These two problems, to the first of which the 27th Prop, is necessary, are the most general and useful of all in the Elements, and are most frequently made use of by the ancient geometers in the solution of other problems ; and tbereft)re are very ignorantly left out by Tacquet and VI. aj] PROPOSITION 27 959 Dechales in their editions of the Elements, who pretend that they are scarce of any use." It is strange that, with this observation before him, even Todhunter should have written as follows. " We have omitted in the sixth Book Propositions 37, z8, 39 and the first solution which Euclid gives of Proposition 30, as they appear now to be never required, and have been condemned as useless t^ various modern commentators ; see Austin, Walker and Lardner." VI. 27 contains the Stapto-ftat, the condition for a real solution, of the problem contained in the proposition following it The maximum of a!l the parallelograms having the given property which can be applied to a given straight line is that which is described upon half the line (to airo t^s ^(luntat ayaypa^/urmr). This corresponds to the condition that an equation of the form u'^ -/jc* = A may have a real root. The correctness of the result may be seen by taking the case in which the parallelograms are rectangles, which enables us to leave out of account the sint of the angle of the parallelograms without any real loss of generality. Suppose the sides of the rect- angle to which the dtfeci is to be similar to be as £ to f, ^ corresponding to the side of the defect which lies along AB. Suppose that AKFG is any parallelogram applied to AB having the given property, that AB=a, and that JFK=x. Then KB = -x, and therefore AK=a- - x. c ( Hence [a-- x\x=-S, where S is the area of the rectangle AKFG. Thus, given the equation F X C K c where 5 is undetermined, vi. J7 tells us that, if x is to have a real value, S cannot be greater than the rectangle CE, Now CB = -, and therefore CD = -..-; a' b 2' whence 5 It i . — , which is just the same result as we obtain by the algebraical method. In the particular case where the defect of the parallelogram is to be t square, (he condition becomes the statement of the fact that, if a straight lint be divided into two parts, the rectangle contained by the parts cannot exceed the squart on half the line. Now suppose that, instead of taking F on BD as in the 6gure of the proposition, we take F on BD produced beyond D but so that DF is less than BD. Complete the figure, as shown, after the manner of the construction in the proposition. afitt) • BOOK VI [vi. 37, 18 Then the pamllelogram FKBH is similar to the given paratlelogram to which the defect is to be simikr. Hence the parallelogram GAKF is also a paralletograin applied to AB and satisfying the given condition. We can now prove that GAKF is less than CE or AD. Let ED produced meet AG in O. Now, since BF is the diagonal of the parallelogram KH, the complements KD, DH are equal. But DH= DG, and DG is greater than OF. Therefore KD > OF. Add OK to each ; and AD, or CE,>AF, • • "■' This other "" case " of the proposition is found in all the MSS,, but Heiberg relegates it to the Appendix as being very obviously interpolated. The reasons for this course are that it is not in Euclid's manner to give a separate demonstration of such a " case "; it is rather his habit to give one case only and to leave the student to satisfy himself about any others (cf. i, 7). Internal evidence is also against the genuineness of the separate proof. It is put after the conclusion of the proposition instead of before it, and, if Euclid had intended to discuss two cases, ne would have distinguished them at the beginning of the proposition, as it was his invariable practice to do. Moreover the second "case" is the less worth giving because it can be so easily reduced to the first. For suppose F' to be taken on BD so that FD - F D. Produce BF to meet AG produced in P. Complete the parallelogram BAPQ, and draw through F" straight lines parallel to and meeting its opposite sides. Then the complement F'Q is equal to the complement AF'. And it is at once seen that AF, F'Q are equal and similar. Hence the solution of the problem represented by AF or F'Q gives a parallelogram of the same size as AF" arrived at as in the first " case." It is worth noting that the actual difference between the parallelogram AF and the maximum area AD that it can possibly have is represented in the figure. The difference is the small parallelc^ram DF. Proposition 28. To a given straight line to apply & parallelogram equal to a given rectilineal figure and deficient by a parallelogrammic figure similar to a. given one : thus the given rectilineal figure must not be greater than the parallelogram, described on the half of the straight line and similar to the defect. Let AB be the given straight line, C the given rectilineal figure to which the figure to be applied to AB is required to be equal, not being greater than the parallelogram described on the half of AB and similar to the defect, and D the parallelogram to which the defect is rt;quired to be similar ; VI. 28] PROPOSITIONS 17, 28 i6t thus it is required to apply to the given straig^^ line AB a parallelogram equal to the given rectilineal figure C and deficient by a parallelogrammic figure which is similar to D, Let ABhe bisected at the point E, and on EB let EBFG be described similar and similarly situated to D ; [vi. 18] let the parallelogram AG he completed. If then AG IS equal to C, that which was enjoined will have been done ; for there has been applied to the given straight line AB the parallelogram AG equal to the given rectilineal figure C and deficient by a parallelogrammic figure GB which is similar to/?. But, if not, let //E be greater than C, Now HE is equal to GB ; therefore GB is also greater than C, Let KLMN be constructed at once equal to the excess by which GB is greater than C and similar and similarly situated to D. [vi, 25} But D is similar to GB \ therefore KM \s also similar to GB. [vi. zi"] Let, then, KL correspond to GE, and LM to GF. Now, since GB is equal to C, KM, therefore GB is greater than KM ; therefore also GE is greater than KL, and GF than LM. Let GO be made equal to KL, and GP equal to LM; and let the parallelogram OCPQ be completed ; therefore it is equal and similar to KM. Therefore GQ is also similar to GB; \y\. i\\ therefore GQ is about the same diameter with GB. [vi. a6] afo r BOOK VI .;t't [vi. aS ,. Let C55 be their diameter, and let the figure be described. Then, since BG is equal to C, KM, and in them GQ is equal to KM, therefore the remainder, the gnomon UWV, is equal to tht remainder C. And, since PR is equal to OS, ' '' let QB L e added to each ; ' • ••* ' • therefore the whole PB is equal to the whole OB. But OB is equal to TE, since the side AE is also equal to the side EB ; [1. 3^] therefore TE is also equal to PB. Let 05 be added to each ; therefore the whole TS is equal to the whole, the gnomon VWU. But the gnomon VWU was proved equal to C ; therefore TS is also equal to C. Therefore to the given straight line AB there has been applied the parallelogram ST equal to the given rectilineal figure Cand deficient by a parallelogrammic figure ^^ which is similar to D. Q. E. F, The second part of the enunciation of this proposition which states the SiapuT/uH appears to have been considerably amplified, but not improved in the process, by Theon. His version would read as follows. " But the given rectilineal figure, that namely to which the applied parallelogmm must be equal (tf S«t lo-ov impa^aXtiv), must not be greater than that applied to the half (irapa/3iiAAo^(Miv instead of avaypa^/inov), the defects being similar, (namely) that (of the parallelogram applied) to the half and that (of the required parallelogram) which must have a similar defect" {onoimr otrui' tuv IWtt/i- fidrwv roD rt air^ rij^ ^fjiurtvK koI cp Sft uftoiov iXKtiTrnv). The first ampHflCfttion "that to which the applied parallelogram must be equal" is quite unnecessary, since " the given rectilineal figure " could mean nothing else. The above attempt at a translation will show how difficult it is to make sense of the words at the end , they speak of two defects apparently and, while one may well be the " defect on the half," the other can hardly be tfu pvm paralUia^am " to which the defect (of the required [^rallelogram) must be similar." Clearly the reading given above (from P) is by far the better. In this proposition and the next there occurs the tacit assumption (already alluded to in the note on vi. 31) that if, of two similar paraiitlogramz, one is gnatfr than the of her, ttther side of the greater is greater tkan the corresponding side of the kit. VI. 98] PROPOSITION sS •*3 As already remarked, vi. 28 is the geometrical equivalent of the solution of the quadratic equation tf jc - - a* • c S. subject to the condition necessary to admit of a real solution, namely that * 4 The corresponding proposition in the Data is (Prop. 58), If a given {ana) be applied (i.e. in the fonn of a parallelogram) to a given straight line and be defieitnt by a figure (i.e. a parallelogram) f<Wfl in species, the breadths of tht defeet are given. To exhibit the exact correspondence between Euclid's geometrical and the ordinary algebraical method of solving the equation we will, as before (in order to avoid bringing in a constant dependent on the sine of the angle of the parallelograms), suppose the parallelograms to be rectangles. To solve the equation algebraically we change the signs and write it -x'-ax = -S. e a* We may now complete the square by adding *'4' Thus ■<Mf + T. — - b 4 e a^ „ and, extracting the square root, we have Now let us observe Euclid's method. - ^ o \ S X s a ■0 He first describes GEBF on EB (half of AB") similar to the given parallelogram D. He then places in one angle FGE of GEBF a similar and similarly situated parallelogram GQ, equal to the difference between the parallelogram GB and the «irea C. With our notation, whence GO: OQ = e:b, » OQ = GO.'^ a64 BOOK VI [vi. 38 Similarly -^ = £B=GE.-, so that GE=i.-. ' a Therefore the parallelogram GQ^ GO* . - , ( a' and the parallelogram GB = 1 • — • Thus, in taking the parallelogram GQ equal to {GB - S), Euclid really finds GO from the equation GO'.''-^i.''--S. f * 4 The value which he finds is ^ and he finds QS (or x) by subiracting GO from GE ; whence ^_t a f'cfe a«^\ It will be observed that Euclid only gives one solution, that corresponding to the negative sign before the radical. But the reason must be the same as that for which he only gives one "case" in vi. s 7. He cannot have failed to see how to adti GO to GE would give another solution. As shown under the last proposition, the other solution can be arrived at (i) by placing the parallelogram GOQ/' in B' A' the angle vertically opposite to FGE so that '. ''--.Q' ^ G(^ lies along BG produced. The parallelo- \ Yx ^' ; \ gram A Q then gives the second solution. The \ I \^ \ \ \ side of this parallelogram lying along AB \% \ \p' x\ '. ' equal to S3. The other side is what we have \ T ' gVv p\ \ called Xy and in this case \ \ \ \^ ', \ x^EG+GO J \ o\ \iQ\ ' * ■ 2 ^ V i (i • 4 -^j ■ S B (z) A parallelogram similar and equal to A<^ can also be obtained by producing BG till it meets A T produced and completing the parallelogram JfABA', whence it is seen that the complement QA' is equal to the comple- ment j4^, besides being equal and sin'ilar and similarly situated to AQ. A particular case of this proposition, indicated in Prop. 85 of the Data, is that in which the sides of the defect are equal, so that the defect is a rhombus with a given angle. Prop. 85 proves that, 1/ two straight lints (oniain a givtn area in a given angle, and the sum of the straight lines be given, each of them E *_ will be gioen also. AB, BC being the / given straight lines "containing a given / area AC in ^ given angle ABC," one / side CB is produced to J) so that BD ^ is equal to AB^ and the parallelograms are completed. Then, by hypothesis, CD is of given length, and .^C is a parallelo- VI. a8, 29] PROPOSITIONS 28, 19 m^; gram applied to CZ> falling short by a rhombus {AD) with a given angle EDB, The case is thus a particular case of Prop. 58 of the Data quoted above (p. 263) as corresponding to vi. 28. A particular case of the last, that namely in which the defect \& a squart, corresponding to the equation is importai.t. This is the problem of applying to a gwen straight lint a tedangli tqual to a given area and falling short by a square ; and it can be solved, without the aid of Book vi., as shown above under 11. 5 (Vol. 1. PP- 383—4). Proposition 29. To a given siraigkt line to apply a parallelogram equal to a given rectilineal figure and exceeding by a parallelogrammic figure similar to a given one. Lei j4B be the given straight line, C the given rectilineal figure to whtcli the figure to be applied to A J} is required to be equal, and D that to which the excess is required to be similar ; thus it is required to apply to the straight line AB a parallelo- gram equal to the rectilineal figure C and exceeding by a parallelogrammic figure similar to D. Let AB be bisected at ^ ; , let there be described on BB the parallelogram B/^ similar and similarly situated to D ; and let G// be constructed at once equal to the sum of BB", C and similar and similarly situated to £f. [vi. 35] Let JC// correspond to BL and /CG to BB. Now, since G// is greater than BB, therefore J^// is also greater than BL, and JCG than BB. SH BOOK VI [vi. 39 Let PL, FE be produced, let FLM be equal to KH, and FEN to KG, and let MN be completed ; therefore MN is both equal and similar to GH, But GH is similar to EL ; therefore ^A?" is also similar to EL ; [vi, a i] therefore J?/, is about the same diameter with MN. [vi. 16] Let their diameter FO be drawn, and let the figure be described. Since GH'\s equal to EL, C, while GH is equal to MN, therefore MN is also equal to EL, C. Let EL be subtracted from each ; therefore the remainder, the gnomon XIVV, is equal to C. Now, since AE is equal to EB, AN is also equal to NB [1. 36], that is, to LP [i. 43]- Let EO be added to each ; therefore the whole AO is equal to the gnomon VIVX, But the gnomon VJVX is equal to C \ therefore AO is also equal to C. Therefore to the given straight line AB there has been applied the parallelogram AO equal to the given rectilineal figure C and exceeding by a parallelogram mic figure QP which is similar to D, since PQ is also similar to EL [vi. n]. Q. E. F. The corresponding proposition in the Data is (Prop. 59), If a given {area) Ar applied (i.e. in the form of a parallelogram) tc a given straight lint exteeding by a fi^re gii'en in species, the breadths of the excess are givi». The problem of vi, 29 corresponds of course to the solution of the quadratic equation ax -y- - 3? = S, c The algebraical solution of this equation gives The exact correspondence of Euclid's method to the aigebraical solution may be seen, as in the case of vi. 28, by supposing the parallelograms to be rectangles. In this case Euclid's construction on EB of the parallelogram EL similar to D is equivalent to finding that FE = \.^-, and EL = \.'^. b z * 4 VI. 19. 3°} PROPOSITIONS 19, 30 *&} His determination of the similar parailelogram />/jV equal to the sum of £L and S corresponds to proving that f d 4 or whence x is found as a Euclid takes, in this case, the solution corresponding to the positive sign before the radical because, from his point of view, that would be the only solution. No iiofiitriiai is necessary because a real geometrical solution is always possible whatever be the size of S. Again the £>afa has a proposition indicating the particular case in which the excess is a rhombus with a given angle. Prop. 84 proves that, 1/ itvo straight lines contain a givtn area in a ^ven angle, and one of thf straight lines is greater than the other by a given straight line, each of the two straight lines is given also. The proof reduces the proposition to a particular case of Data, Prop. 59, quoted above aa corresponding to vi. 29. Again there is an important particular case which can be solved by means of Book II. only, as shown under 11. 6 above {Vol. 1. pp. 386 — 8), the case namely in which the excess is a square, corresponding to the solution of the equation This is the problem of applying to a given straight line a rectangle equal to a ffven area and exceeding by a square. Proposition 30. To cut a given finite straight line in extreme and mean ratio. Let AB be the given finite straight line ; thus it is required to cut AB in extreme and mean ratio. On AB let the square BC be described ; and let there be applied to AC the parallelo- gram CD equal to BC and exceeding by the figure AD similar to BC. [vi. 29] Now BC is a square ; therefore AD is also a square. And, since BC is equal to CD, let CE be subtracted from each ; therefore the remainder BF is equal to the remainder AD. •88 cf , BOOK VI [vi. 30, 31 But it is also equiangular with it ; therefore in BF, AD the sides about the equal angles are reciprocally proportional ; [vi- h] therefore, as FE is to ED, so is AE to EB, But FE is equal to AB, and ED to AE, Therefore, as BA is to AE, so is AE to EB. And AB is greater than AE ; therefore AE is also greater than EB. Therefore the straight line AB has been cut in extreme and mean ratio at E, and the greater segment of it is AE. Q. E. F. . It will be observed that the construction in the tent is a direct application of the preceding Prop. 29 in the particular case where the txass of the parallel ogram which is applied is a squar*. This fact coupled with the position of V(, 30 is a sufficient indication that the construction is Euclid's. In one place Theon appears to have amplified the argument. The text above says "But fE is equal to AB" while the mss. B, F, V and p have " But /^ff is equal to ^ C, that is, to ^^." The MSS. give after ^vtp Ihu iroujiriH an alternative construction which Heibei^ relegates to the Appendix. Thf text-books give this construction alone and leave out the other. It will be remembered that the alternative proof does no more than refer to the equivalent construction in ii. 11. "Let AB \ie. cut at C so that the rectangle AB, BC is equal to the square on CA. [ii. ( jj Since then the rectangle AB, BC is equal to the stjuare on CA, therefore, as BA is to .rfC, so is /4Cto CB. [vi. 17] Therefore AB has been cut in extreme and mean ratio at tV It is intrinsically improbable that this alternative construction was added to the other by Euclid himself. It is however just the kind of interpolation that might be expected from an editor. If Euclid had preferred the alternative construction, he would have been more likely to give it alone. I '■ Proposition 31. In right-angled triangles the figure on Ike side suit ending the right angle is equal to the similar and similarly described Jtgures on the sides containing the right angle. L&t ABC be a right-angled triangle having the angle BAC right ; I say that the figure on BC is equal to the similar and similarly described figures on BA, AC. Let AD be drawn perpendicular. Then since, in the right-angled triangle ABC, AD has Vf. 3'] PROPOSITIONS 30, 31 been drawn from the right angle at A perpendicular to the base BC, the triangles ABD, v4/?C adjoin- ing the perpendicular are similar both to the whole ABC and to one another, [vi, 8] And, since ABC is similar to ABD, therefore, as CB is to BA^ so is AB to BD. [vi. I>er. 1] And, since three straight lines are proportional, as the first is to the third, so is the figure on the first to the similar and similarly described figure on the second, [vi. 19, For.] Ihercfore, as CB is to BD, so is the figure on CB to the similar and similarly described figure on BA. .-•■• ■ ^ 'l^ '. For the same reason also, ' as BC is to CD, so is the figure on BC to that on CA ; so that, in addition, as BC is to BD, DC, so is the figure on BC to the similar and similarly described figures on BA, AC. But BC is equal to BD, DC ; therefore the figure on BC is also equal to the similar and similarly described figures on BA, AC. Therefore etc. Q. E. D. As we have seen (note on i. 47), this extension of [. 4; is credited by Proclus to Euclid persor\ally. There is one inference in the proof which requires examination. Euchd proves that CB : .ff/> = (figure on CB) -. (figure on BA), and that BC : CZ> = (figure on BC) : (figure on CA), and then infers directly that BC : {BD+ CZ))- (fig. on BC) : (sum of figs, on BA and AC). Apparently v. 24 must be relied on ss justifying this inference. But it is not directly applicable ; for what it proves is that, if a:b = c:d, .,-..• A-^ >.V-. and ■ f:b=f:d, • .-'M^V then (a + (t) : b = {f \f) : d. Thus we should itwtrt the first two proportions given above (by Simson's »jo «t . BOOK VI [vi. 31, 31 Prop. B which, as we have seen, is a direct consequence of the definition of proportion), and thence infer by v. 34 that {BD+CD) : J3C= {sum of figs, on £j4, AC) : (fig. on BC). But BD 4 CD is equal to BC; therefore (by Simson's Prop. A, which again is an immediate consequence of the definition of proportion) the sum of the figures on BA, AC is equal to the figure on BC. The Mss. ag^in give an alternative proof which Heibetg places in the Appendix. It first shows that the simitar figures on the three sides have the same ratios to one another as the squara on the sides respectively. Whence, by using I. 47 and the same argument based on v. 24 as that explained above, the result is obtained. If it is considered essential to have a proof which does not use Simson's Props, fi and A or any proposition but those actually given by Euclid, no method occurs to me except the following. Eucl. V. 12 proves that, if a, J, c are three magnitudes, and d, e, f three Others, such that a : b=:d : t, • •..-:- :.: ■ bie'-f.f, • ., : then, ex aeptali, a : c = d :/, , ' i ' If now in addition a •.b=b -.Cf so that, also, d i e = e :/, the ratio a -. ( ii duplicate of the ratio a : i, and the ratio d :/ duplicate of the ratio d : t, whence the ratios which ar< duplicate of equal ratios are equal. Now (fig. on AC)\ (fig. on AB) = the ratio duplicate of AC : AB = the ratio duplicate of CD : DA = CD : BD. H»ice (sum of figs, on AC, AB) : (fig. on AB) = BC -. BD. [v. iS] But ' (fig. on BC) : (fig. on AB) = BC : BD (as in Euclid's proof). Therefore the sum of the figures on AC, AB has to the figure on AB the same ratio as the figure on BC has to the figure on AB, whence the figures on AC, AB are together equal to the figure on BC- [v. 9] Proposition 32. If two triangles having two sides proportional to two sides be placed together at one angle so that their corresponding sides are also parallel, the remaining sides of the triangles will be in a straight line. Let ABC, DCE be two triangles having the two sides BA, AC proportional to the two sides DC, DE, so that, as AB is to A C, so is DC to D£, and AB parallel to DC, and AC to D£; I say that BC is in a straight line with C£. yi. 3a] PROPOSITIONS 31, 3a ^1 For, since AB is parallel to DC, and the straight line AC has fallen upon them, the alternate angles BAC, ACD are equal to one another, [r. 19] For the same reason the angle CDE is also equal to the angle A CD ; so that the angle BAC is equal to the angle CDE. And, since ABC, DCE are two triangles having one angle, the angle at A, equal to one angle, the angle at D, and the sides about the equal angles proportional, so that, as BA \%\ja AC, so is CD to DE, therefore the triangle ABC is equiangular with the triangle DCE ; [vi. 6] therefore the angle ABC is equal to the angle DCE. But the angle ^CZ? was also proved equal to the angle BAC\ therefore the whole angle A CE is equal to the two angles ABC, BAC. Let the angle ACB be added to each ; therefore the angles ACE, ACB are equal to the angles BAC, ACB, CBA. But the angles BA C, ABC, A CB are equal to two right angles ; [1. 3a] therefore the angles ACE, ACB are also equal to two right angles. Therefore with a straight line AC, and at the point C on it, the two straight lines BC, CE not lying on the same side make the adjacent angles ACE, ACB equal to two right angles ; therefore BC is in a straight line with CE. [1. 14] Therefore etc. Q. E. D. It has often been pointed out (e.g. by Clavm&, Lardner and Todhunter) that the enunciation of this proposition is not precise enough. Suppose that ijt BOOK VI ' [vi. 32 ABC is a triangle. From C draw CD parallel to BA and of any length. From D draw D£ parallel to CA and of such length that CD:DE = BA : AC. Then the triangles ABC, BCD, which have the angular jxiint C coniRion literally satisfy Euclid's enunciation ; but by no possi- bility can CE be iti a straight line with CB if, as in the case supposed, the angles included by the corresponding sides are supplementary (unless both are right angles). Hence the included angles must be equal, so that the triangles must be similar. That being so, if they are £0 have nothing more than one angular point common, and two pairs of corresponding sides are to he/aralMas distinguished froni one or both being in the same slraigiU line, the triangles can only be placed so that the corresponding sides in both are on the same side of the third side of either, and the sides (other than the third sides) which meet at the common angular point are not corre- sponding sides. Todhunter remarks that the proposition seems of no use. Presumably he did not know that it is used by Euclid himself in xiii. 17. This is so however, and therefore it was not necessary, as several writers have thought, to do away with the proposition and Irnd a substitute which should be more useful. 1. De Morgan proposes this theorem : "If two similar triangles be placed with their bases parallel, and the equal angles at the bases towards the same parts, the other sides are parallel, each to each ; or one pair of sides are in the same straight line and the other pair are parallel," 2. Ur Lachlan substitutes the somewhat similar theorem, "If two similar triangles be placed so that two sides of the one are parallel to the corresponding q sides of the other, the third sides are a /\ parallel," /\ / \ But it is to be observed that these qZ.....,\.. Z. a propositions can be proved without / \ using Book vi. at all ; they can be / \ proved from Book i,, and the triangles = c may as well be called "equiangular" simply. It is true that Book vi. is no more than formally nece^ary to Euchd's proposition. He merely uses vt, 6 because his enunciation does not say that the triangles are similar ; and he only proves them to be similar in order to conclude that they are equiangular. From this point of view Mr Taylor's substitute seems the best, viz, 3. "If two triangles have sides parallel in pairs, the straight lines joining the corresponding vertices meet in a point, or are parallel." Simson has a theory (unnecessary in the circumstances) as to the possible object of VI. 32 as it stands. He points out that the enunciation of vi, 26 might be more general so as to cover the case of similar and similarly situated parallelo- grams with equal angles not coincident but vertically opposite. It can then be proved that the diagonals drawn VI- 3S. 33] PROPOSITIONS 32, 33 ns through the common angular point are In one straight line. If ABCF, CDEG be similar and simikrl)' situated parallelograms, so that BCGf DCF are straight lines, and if the diagonals AC^ CE be drawn, the triangles ABC, CDR are similar and are plated exactly as deseriied in vi, 32, so that AQ CE are in a straight line. Hence Sim son suggests that there may have been, in addition to the in- direct demonstration in vi. 26, a direct proof covering the case just given which may have used the result of vi, 31. I think however that the place given to the latter proposition in Book vi. is against this view. Proposition 33. In equal circles angles have the same ratio as the circum- ferences on which they stand, whether they stand at the centres or at the circumferences. Let ABC, DEF be equal circles, and let the angles BGC, EHFhG. angles at their centres G, H, and the angles BAC, £'/?/^ angles at the circumferences ; I say that, as the circumference BC is to the circumference £F, so is the angle BGC to the angle EHF, and the angle ^^C to the angle jp/?/^. For let any number of consecutive circumferences CK, KL be made equal to the circumference BC, and any number of consecutive circumferences FM, MN equal to the circumference EF', and let GK, GL, HM, HN be joined. Then, since the circumferences BC, CK, KL are equal to one another, the angles BGC, CGK, KGL are also equal to one another ; [in. 27] i14 BOOK VI ' [vi. 33 therefore, whatever multiple the circumference BL is of BC, that multiple also is the angle BGL of the angle BGC. For the same reason also, whatever multiple the circumference NE is of EF, that multiple also is the angle NHE of the angle EHF. If then the circumference BL is equal to the circumference ^A^, the angle BGL is also equal to the ^ug\&EHN; [in. ij] if the circumference BL is greater than the circumference EN, the angle BGL is also greater than the angle EHN ; and, if less, less. There being then four magnitudes, two circumferences BC, EF, and two angles BGC, EHF, there have been taken, of the circumference BC and the angle BGC equimultiples, namely the circumference BL and the angle BGL, and of the circumference EF and the angle EHF equi- multiples, namely the circumference EN and the angle EHN. And it has been proved that, if the circumference BL is in excess of the circumference EN, the angle BGL is also in excess of the angle EHN ; if equal, equal ; and if less, less. Therefore, as the circumference BC is to EF, so is the angle BGC to the angle EHF. [v. Def. 5] But, as the angle BGC is to the angle EHF, so is the angle BAC to the angle EDF; for they are doubles respec- tively. Therefore also, as the circumference BC is to the circum- ference EF, so is the angle BGC to the angle EHF, and the angle BA C to the angle EDF. Therefore etc, '■ ■ Q. E. D. This proposition as generally given includes a second part relating to sectors of circles, corresponding to the following words addt^ to the enunciation : " and further the sectors, as constructed at the centres " {hi it not al to/«« art [or oiTt] irf)o« rots «rKTpo« ot)hotq'^(voi). There is of course a corresponding addition to the "definition" or "particular statement," "and further the sector GBOC to the sector HEQF" These additions are clearly due to Theon, as may be gathered from his own statenient in his commentary on the ^Sij^tijoJ <ru\na^i<i of PtoIemy, " But that sectors in equal circles are to one another as the angles on which they stand, has been proved by me in my edition of the VI. 33] PROPOSITION 33 -^S Elements at the end of the sixth book." Campanus omits them, and P has them only in a later hand in the margin or between the lines. Theon's proof scarcely needs to be given here in full, as it can easily be supplied. From the equality of the arcs BC, CK he infers [in. 29] the equality of the chords BC, CK, Hence, the radii being equal, the triangles GBC, GCK are equal in all respects [i. 8, 4]. Next, since the arcs BC, CK are equal, so are Xhz arcs BAC, CAK. Therefore the angles at the circumference subtended by the latter, i.e. the angles in the segments BOC, CPK, are equal [iii. v\\ and the segments are therefore similar [in. Def.. iil and equal [in. 24]. Adding to the equal segments the equal tnangles GBC^ CCA" respectively, we see that the sectors GBC, GCK are equal. Thus, in equal circles, sectors standing on equal arcs are equal ; and the rest of the proof proceeds as in Euclid's proposition. As regards Euclid's proposition itself, it will be noted that (i), besides quoting the theorem in hi. 27 that in equal circles angles which stand on equal arcs are equal, the proof assumes that the angle standing on a greater arc is greater and that standing on a less arc is less. This is indeed a suffi- ciently obvious deduction from in. 27. (2) Any equimultiples whaievtr are taken of the angle BGC and the arc BC, and any equimultiples whatever of the angle EHF and the arc EF. (Accordingly the words "any (quimuUipUs whaievtr" should have been used in the step immediately preceding the inference that the angles are proportional to the arcs, where the text merely states that there have been taken of the circumference BC and the angle BGC equimultiples BL and BGL.) But, if any multipk of an angle is regarded as being itself an angle, it follows that the restriction in 1. Deff, 8, 10, ii, 12 of the term angltio an angle less than two right angles is implicitly given up ; as De Morgan says, "the angle breaks prison," Mr Dodgson {Euclid and his Modern Rivals, p, 193} argues that Euclid conceived of the multiple of an angle as so many separate angles not added together into one, and that, when it is inferred that, where two such multiples of an angle are equal, the arcs subtended are also equal, the argu- ment is that the sum total of the first set of angles is equal to the sum total of the second set, and hence the second set can be broken up and put together again in such amounts as to make a set equal, each to each, to the first set, and then the sum total of the arcs will evidently be equal also. If on the other hand the multiples of the angles are regarded as single angular magnitudes, the equality of the subtending arcs is not inferrible directly from Euclid, because his proof of ni. 26 only applies to cases where the angle is less than the sum of two right angles, (.'^s a matter of fact, it is a question of inferring equality of angles or multiples of angles from equality of arcs, and not the converse, so that the reference should have been to in. 27, but this does not affect the question at issue.) Of course it is against this view of Mr Dodgson that Euclid speaks throughout of " the angle BGL " and " the an^e EHN " (ij imo BHA ■yiui'i'a, jj ujro E0N yoivia). I think the probable explanation is that here, as in in. 20, 21, 26 and 27, Euclid deliberately took no cognisance of the case in which the multiples of the angles in question would be greater than two right angles. If his attention had been called to the fact that in. 20 takes no account of the case where the segment is less than a semicircle, so that the angle in the segment is obtuse, and therefore the " angle at the centre " in that case (if the term were still applicable) would be »j« BOOK VI [vi. 33 greater than two right angles, Euclid would no doubt have refused to regard the latter as an angle, and would have represented it otherwise, e.g. as the sum of two angles or e^ what is left when an angU in the true sense is sub- tracted from four right angles. Here then, if Euclid had been asked what course he would take if the multiples of the angles in question should be greater than two right angles, he would probably have represented them, I think, as being equal to so many right anghs plus an angU less than a right angle, or so many limes two right angles plus an angle, acute or obtuse. Then the equality of the arcs would be the equality of the sums of so many circum- ferences, semi-circumferences or quadrants plus arcs less than a semicircle or a quadrant. Hence I agree with Mr Uodgson that vi. 33 affords no evidence of a recognition by Euclid of " angles " greater than two right angles Theon adds to his theorem about sectors the Porism that. As the sector is to the sector, so also is the angle to the angle. This corollary was used by Zenodorus in his tract -n-fpt uro/ifrpcuv ax^p-ituiv preserved by Theon in his commentar)' on Ptolemy's oi/n-afu, unless indeed 'i'heon himself interpolated the words {^s S" tom^ut tt/m^ T^tv rofiia, tJ vro E0A yiui'Ja Trp^ t^v viro M@A), .1 ,1. BOOK VII. DEFINITIONS. 1. An unit is that by virtue of which each of the things that exist is called one. • ' 2. A number is a multitude composed of units. 3. A number is a part of a number, the less of the greater, when it measures the greater ; 4. but parts when it does not measure it. 5. Tue greater number is a multiple of the less when it is measured by the less. 6. An even number is that which is divisible into two equal parts, 7. An odd number is that which is not divisible into two equal parts, or that which differs by an unit from an even number, ■ . ... 8. An even-times even number is that which is measured by an even number according to an even number. 9. An even-times odd number is that which is measured by an even number according to an odd number, 10. An odd-times odd number is that which is measured by an odd number according to an odd number. gft "■' BOOK VII [vii. DEFF, II — as 11. A prime number is that which is measured by an unit alone, .. ^ . 12. Numbers prime to one another are those which are measured by an unit alone as a common measure. 13. A composite number is that which is measured by some number. 14. Numbers composite to one another are those which are measured by some number as a common measure. 15. A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced. 16. And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another. 17. And, when three numbers having multiplied one another make some number, the number so produced is solid, and its sides are the numbers which have multiplied one another. 18. A square number is equal multiplied by equal, or a number which is contained by two equal numbers. 19. And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers. 20. Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth. 21. Similar plane and solid numbers are those which have their sides proportional. 22. A perfect number is that which is equal to its own parts. VII DM. i] DEFINITIONS <- 3>9 Definition i. Movas teric, Kaff 4jv inatrrov tav 6vTtov ty Xryerai. lamblichus (fl. ana 300 a.d.) ceKs us (Comm. on Nieemaehus, ed. Pistelli, p. 1 1, 5) that the Euclidean definition of an unii or a monad was the definition given by " more recent " writers (ol vta/rtpoi.), and that it lacked the words "even though it be collective" {ni-v (twmiiijiTuthi %). He also gives (ibid. p. 11) a number of other definitions, (t) According to "some of the Pytha- goreans," " an unit is the boundary between number and parts " (^omt brrar ApSfuiv KQi noftiuai iitSopuir), " because from it, as from a seed and eternal root, ratios increase reciprocally on either side," i.e. on one side we have multiple ratios continually increasing and on the other (if the unit be sub- divided) submultiple ratios with denominators continually increasing. (2) A somewhat similar definition is that of Thyniaridas, an ancient Pyth^orean, who defined a monad as " limiting quantity " (ir«patVow<ra iroowij!), the beginning and the end of a thing being equally an extremity {vipa.%). Perhaps the words together with their explanation may hest be expressed by " limit of fewness." Theon of Smyrna (p. 18, 6, ed. Hill'er) adds the explanation that the monad is " that which, when the multitude ts diminished by way of continued subtraction, is deprived of all number and takes an abiding position {liBv^) and rest." If, after arriving at an unit in this way, we proceed to divide the unit itself into parts, we straightway have multitude again. (3) Some, ac- cording to lamblichus (p. r r, 16), defined it as the "form of forms" (dStuK <I8«) because it potentially comprehends all forms of number, e.g. it is a polygonal number of any number of sides from three upwards, a solid number in all forms, and so on, (We are forcibly reminded of the latest theories of number as a "Gattung" of "Mengen" or as a "class of classes.") {4) Again an unit, says lamblichus, is the first, or smallest, in the category of how many (rixrov), the common part or beginning of Aow many. Aristotle defines it as " the indivisible in the (category of) quantity," to xari to voaov dSuupfrov {Mdaph. 10S9 b 35), ttottoy including in Aristotle continuous as well as discrete quantity ; hence it is distinguished from a point by the fact that it has not position : "Of the indivisible in the category of, and quA^ quantity, that which is every way (indivisible) and destitute of position is called an unii, and that which is every way indivisible and has position is a point" (Meiaph. io[6b2S). (5) In accordance with the last distinction, Aristotle calls the unit " a point without position," ariyiii) afftros {Mtlaph. 1084 b 26), (6) Lastly, lamblichus says that the school of Chrysippus defined it in a con- fused manner {<niyKr)(V)Uviiii) as " multitude one {ttk^Om tv)," whereas it is alone contrasted with multitude. On a comparison of these definitions, it would seem that Euclid intended his to be a more popular one than those of his predecessors, S);/ui!%c, as Nicomachus called Euclid's definition of an iven number. The etymological signification of the word floras is supposed by Theon of Smyrna (p. 19, 7 — 13) to be either (i) that it remains unaltered if it be multiplied by itself any number of times, or (») that it is separated and isolattd (^MjaofiHa^at) from the rest of the multitude of numbers. Nicomachus also observes (1. 8, a) that, while any number is half the sum (i) of the adjacent numbers on each side, (2) of numbers equidistant on each side, the unit is mo$i solitary (/iwdtran;) in that it has not a number on each side but only on one side, and it is half of the latter done, i.e. of 2. aSO BOOK VII [Vll, DEFF, 2—4 Definition 2. The definition of a numbir is again only one out ol many that are on record. Nicomachus (i. 7, i) combines several into one, saying that it is " a defined multitude (ttXij^o? (ijptcr^«Vor), or a collection of unit.s (^afa£ti>v o-uo-ni^ta), or a flow of quantity made Up of units " (ttoitotiitik x^fo *" liova&aiv avyKtljufvov). Theon, in words almost identical with those attributed by Stobaeus {Edogae, 1. i, 8) to Moderatus, a Pythagorean, says (p. 18, 3—5): " A number is a collection of units, or a progression (jrpcHro&trfio'v) of mul- titude beginning from an unit and a retrogression (dva7ro8«rfio'«) ceasing at an unit." According to lamblichus (p. 10) the description "collection of units" (^ovaSujv o-u'crnj/ui) was applied to the how many, i.e. to number, by Thales, following the Egyptian view (kq™ to \\.yvimaKov aftioKov), while it was Eudoxus the Pythagorean who said that a number was "a defined multitude" (n-Xijfio! Kifn.tr \i.ivav). Aristotle has a number of definitions which come to the same thing: "limited multitude" (irX^flos- to TrfTrfpao-^tfof, Metapk. lOio a 13), "multitude" (or "combination") "of units" or "multitude of indivi- sibles" (ibid. 1 05 J a 30, 1039 a 12, 1085 b 22), "several outi" i^va. TrXtim, Phys. HI. 7, 207 b 7), "multitude measurable by one" {Meiaph. 1057 a 3) and " multitude measured and multitude of measures," the " measure " being unity, TO tv {ibid. 1088 a 5). Definition 3. By a pari Euclid means a submultiple, as he does in v. Def. i, with which definition this one is identical except for the substitution of number (aptSyaojJ for magnitude (/tiyfffo^) ; cf note on v. Def. 1 , Nicomachus uses the word "submultiple" (vwmroXKaTrXairwi) also. He defines it in a way corre.'jponding to his definition of multiple (see note on Def 5 below) as follows (1, 18, 2): " The submultiple, which is by nature first in the division of inequality (called) less, is the number which, when compared with a greater, can measure it more times than once so as to fill it exactly (jrAj^poufru^)." Simi- larly sub-double (iIiroStTrXao-io;) is found in Nicomachus meaning half, and so on. Definition 4. Mept) S{, orav /i^ KaTO^tTp^. By the expression parts {fiipij, the plural of ftipirt) Euclid denotes what we should call A proper fraition. That is, a pari being a submultiple, the rather inconvenient term parts means any numbtrr of such submultiples making up a fraction less than unity. I have not, found the word used in this special sense elsewhere, e.g. in Nicomachus, Theon of Srnyrna or lamblichus, except in one place of Theon (p. 79, 26) where it is used of a proper fraction, of which 1^ is an illustration. viu OBFF. s— 8] NOTES ON DEFINITIONS 2—8 381 Definition $. The definition of a multiple is identical with that in v. Def, 2, except that the masculine of the adjectives is used agreeing with npi^pi's understood instead of the neuter agreeing with fitytOo^ understood. Nicomachus (i. 18, i) defines a multiple as being "a species of the greater which is naturally- first in order and origin, being the number which, when considered in com- parison with another, contains it in itself completely more than once." Definitions 6, 7, 6. ApniK apt6)iot ivTiv 6 Sijfa Siaipovittro<,. 7. Tlfpicro'os oc fLrj oiatfjo^fi.H'Oi ^t)^a. ^ [o] ftovd^i Sia^pbrv Aprlov &pi$fLOv. Nicomachus (1. 7, a) somewhat amplifies these definitions of eve/t and oild numbers thus, "That is evtri which is capable of being divided into two equal parts without an unit falling in the middle, and that is odd which cannot be divided into two equal parts because of the aforesaid intervention (/i«ri- niay) of the unit." He adds that this definition is derived " from the popular conception " {Ik tij! Sij/iuBout uToAij^fon). In contrast to this, he gives (t. 7, 3) the Pythagorean definition, which is, as usual, interesting. "An av» number is that which admits of being divided, by one and the same operation, into the greatest and the least (parts), greatest in size (infKtKo-njTi) hut least in quantity (irmroDTTt). ..while an odd number is that which cannot be so treated, but is divided into two unequal parts." That is, as lamblichus says (p. 12, 2—9), an even number is divided into parts which are the p-e<iUst possible "parts," namely halves, and into the ftwtst possible, namely two, two being the first " num- ber" or "collection of units." According to another ancient definition quoted by Nicomachus (i. 7, 4), an even number is that which can be divided both into two equal parts and into two unequal parts {except the first one, the number 2, which is only susceptible of division into eiiuals), but, however it is divided, must have its two parts of the same kind, i.e. both even or both odd ; while an odd number is that which can only l)e divided into two unequal parts, and those parts always of differait kinds, i.e. one odd and one even. Lastly, the definition of odd and even "by means of each other" says that an odd number is that which differs by an unit from an even number on both sides of it, and an even number that which differs by an unit from an odd number on each side. This alternative definition of an odd number is the same thing as the second half of Euclid's definition, " the number which differs by an unit from an even number." This evidently pre-Euclidean definition is condemned by Aristotle as unscientific, because odd and even are coordinate, both being differentiae of number, so that one should not be defined by means of the other {Topics vi. 4, 142 b 7 — 10). Definition 8. ^Kpnaxi^ aprio^ aptVfjLO^ temv o vtto ofyrlov dptBfAOv fitrpovfitvo^ Kara apTiOV iipi$fi6v. Euclid's definition of an ei'en-times even number dift'ers from that given by the later writers, Nicomachus, Theon of Smyrna and lamblichus ; and the inconvenience of it is shown when we come to ix. 34, where tt is proved iSi BOOK Vn '• [vit. DCFF. 8, 9 ihat A. certain sort of number is Aa/A "even-times even "and "even-times odd." According to the more precise classification of the three other authorities, the " even-times even " and the " even-times odd " are mutually exclusive and are two of three subdivisions into which even numbers fall. Of these three sub- divisions the "even-times evt;ii " and the "tven-timcs odd" form the extremes, and the "odd-times even" is as it were intermediate, showing the character of both extremes (cf, note on the following definition). The even-times emn is then the number which has its halves even, the halves of the halves even, and so on, until unity is reached. In short the evtn-times evin number is always of the form z". Hence lamblichus (pp. lo, zi) says Euclid's definition of it as that which is measured by an even number an even number of times is erroneous. In support of this he quotes the numljer 24 which is four times 6, or six times 4, but yet is not " even times even " according to Euclid himself (oiSi na/ auToi'), by which he musjl apparently mean that 24 is also 8 times 3, which does not satisfy Euclid's definition, '['here can however be no doubt that Euclid meant what he said in his definition as wt have it ; otherwise ix. 32, which proves that a number of the form 2" is even-limes even only, would be quite superfluous and a mere repetition of the definition, while, as already stated, IX. 34 clearly indicates Euclid's view- that a numlier might at the same time be both even. times even and even-times odd. Hence the ftdt'ias which some editor of the commentary of I'hiloponus on Nicomachus found in some copies, making the definition say that the even-times even number is only measured by even numbers an even number of times, is evidently an interpo- lation by some one who wished to reconcile Euclid's definition with the Pythagorean (cf Heiberg, Eiiklid-siudien, p. a 00). A consequential characteristic of the series of even-times even numbers noted by Nicomachus brings in a curious use of the word Suvq^i (generally power in the sense of square, or square root). He says (). 8, 6 — 7) that any part, i.e. any submultiple, of an even -times even number is called by an even- times even designation, while it also has an even-times even value {it is apTtciitts dpTmSurafioi'} when expressed as so many actual units. That is, the -,th part of 2" (where m is less than «) is called after the even-times even number z™, while its actual value (Sufa/ut) in units is 2"-"*, which is also an even-times even number. Thus all the parts, or submultiples, of even-times even numbers, as well as the even-times even numbers themselves, are con- nected with one kind of number only, the even. Definition 9. 'ApTiaKit S{ TTtpunrd; hrtw h viro iptiov ifuB^oxt fLcrpovjucvos Kara npurvvi' Euclid uses the term even- times odd (dpTniic« vipvaao^), whereas Nicomachus and the others make it one word, even-odd (aprtairifiirro^). According to the stricter definition given by the latter (i, 9, i), the even-odd number is related to the even-limes even as the other extreme. It is such a number as, when once halved, leaves as quotient an odd number; that is, it is of the form j(2«+ i). Nicomachus sets the even-odd numbers out as follows, 6, 10, 14, 18, zz, 26, 30, etc. In this case, as Nicomachus observes, any part, or submultipie, is called by a name not corresponding in kind to its actual value (Sura/iw) in units. Thus, viL DEF. 9] NOTES ON DEFINITIONS 8, 9 183 in the case of i3, the ^ part is calied a.rter the even number 2, but its va/ue is the odd number 9, and the Jrd part is called after the odd number 3, while its value is the even number 6, and so on. The third class of even numbers according to the strict subdivision is the odd-even {TrtpunrdpTioi). Numbers are of this class when they can be halved twice or more times successively, but the quotient left when they can no longer be halved is an odd number and not unity. They are therefore of the form 2"'^'{7« + 1), where », nt are integers. They are, so to say, inter- mediate between, or a mixture of, the extreme classes eiien-times tvin and even- odd, for the following reasons, (r) Their subdivision by 3 proceeds for some way like that of the even-times even, but ends in the way that the division of the even-odd by i ends. (2) The numbers after which submultiples are called and their value {Sv'raj««) in units may be both of one kind, i.e. both odd or both even {as in the case of the even-times even), or again may be one odd and one even as in the case of the even-odd. For example »4 is an odd-even number; the ^ ih, tV'^'i ir'h or ^ parts of it are even, but the Jrd part of it, or 8, is even, and the Jth part of it, or 3, is odd. (3) "Nicomachus shows (i. 10, 6 — 9) how to form all the numbers of the odd-even class. Set out two lines (a) of odd numbers beginning with 3, {fi) of even-times even numbers beginning with 4, thus : (a) 3. S> 7i 9i r»> »3. >S et_c. {b) 4, 8, 16, 3*, 64, 128, 156 etc. Now multiply each of the first numbers into each df the second row. Let the products of one of the first into all the second set make horizontal rows ; we then get the rows 12,24, 48, 96,192, 384, 768 etc. 20, 40, 80, 160, 320, 640, 1280 etc, 28, 56, 1 1 J, i24, 448, 896, 179J etc. ,•- 36, 72, 144, 288, 576, 1152, 2304 etc, and so on. Now, says Nicomachus, you will be surprised to see (^(Tfcrtrat trm ftiupMr- rwi) that (a) the vertical rows have the property of the ezfen-odd series, 6, 10, 14, 18, 22 etc., viz, that, if an odd number of successive numbers be taken, the middle number is half the sum of the extremes, and if an even number, the two middle numbers together are equal to the sum of the extremes, (t) the horizontal rows have the property of the even-times even series 4, 8, 16 etc., viz. that the product of the extremes of any number of successive terms is equal, if their number be odd, to the square of the middle term, or, if their number be even, to the product of the two middle terms. Let us now return to Euclid. His 9th definition states that an even-timef odd number is a number which, when divided by an even number, gives an odd number as quotient. Following this definition in our text comes a loth definition which defines an odd-times even number ; this is stated to be a number which, when divided by an odd number, gives an even number as quotient. According to these definitions any even-times odd number would also be odd-times even, anti, from the fact that lamblichus notes this, we may fairly conclude that he found Def, 10 as well as Def. 9 in the text of Euclid which he used. But, if both definitions are genuine, the erjunciations of ix. 33 and IX. 34 as we have them present difficulties, ix. 33 says that " If a num- ber have its half odd, it is even-times odd only " ; but, on the assumption that ^ . ,; BOOK VII -,!• [vir. DKKf. 9— 11 both definitions are genuine, Ihis would not be true, for the number would be odd-timts even as well. ix. 34 says that " If a number neither be one of those which are continually doubled from 2, nor have its half odd, it is both even- times even and even- times odd." The term odd-timts even {irtpuraaKK aprtot) not occurring in these propositions, nor anywhere else after the definition, that definition liecomes superfluous. Iambi ichus however {p. 24, 7 — m) quotes these enunciations differently. In the first he has instead of " even-times odd only " the words " both tven-timei odd and odd-times even " ; and, in the second, for " both even-times even and even-times odd " he has " is both even-times even and at the same time even-times odd and odd-times even." In both cases therefore " odd-times even " is added to the enunciation as lamblichus had it^ the words catrnot have been added by lamblichus himself because he himself does not use the term odd-timts even, but the one word odd-even (rtfiuTirafyno^). In Order to get over the difficulties involved by Def, 10 and these differences of reading we have practically to choose between (i) accept- ing lamblichus' reading in all three places and (2) adhering to the reading of our Mss. in ix. 33, 34 and rejecting Def. 10 altogether as an interpolation. Now the readings of our text of ix. 33, 34 are those of the Vatican MS. and the Theonine mss. as well ; hence they must go back to a time before Theon, and must therefore be almost as old as those of lamblichus. Heiberg considers it improbable that Euclid would wish to maintain a point- less distinction between even-times odd and odd-times even, and on the whole concludes that IJef. [O was first interpolated by some ignorant person who did not notice the difference between the Euclidean and Pythagorean clissi- fication, but merely noticed the absence of a definition of odd-times even and fabricated one as a companion to the other. When this was done, it would be easy to see that the statement in )X. 33 that the number referred to is " even-times odd only " was not strictly true, and that the addition of the words "and odd-times even" was necessary in ix. 33 and tx. 34 as well. Definition 10, n*p«r<raKW Be letpuruoi apSjtan iarw viro irtpurami dpiff/iov fierpois/it™? Kara Trtpt(r<rov dpiOfxov* The Olid-times add number is not defined as such by Nicomachus and lamblichus ; for them these numbers would apparently belong to the «»«- ^osite subdivision of odd numbers. Theon of Smyrna on the other hand says (p. 23, 21) that odd-times odd was one of the names applied to prime numbers (excluding 2), for these have two odd factors, namely i and the number itself. This is certainly a curious use of the term. Definition ii. IXpitfTiK d,pSiia% ttTTiv b /tovaSi /lovij ftfrpttijitvov. A prijne number {vfiS/ro^ iptSjtios) is called by Nicomachus, Theon, and lamblichus a " prime irnrf inwm/oj(V^ (iovk^trtre) number." Theon (p. 23, 9) defines it practically as Euclid does, viz. as a number "measured by no number, but by an unit only." Aristotle too says that a prime number is not measured by any number (Ana/, post. ti. 13, 96 a 36), an unit not being a number [Metaph, 1088 a 5), but only the beginning of number (Theon of Smyrna says the same thii^, p. t\, 13). According to Nicomachus (1. 11, a) the prime number is a VII. DEFK. [I, 12] NOTES ON DEFINITIONS 9—12 285 subdivision, not of numbers, but of odd numbers; it is "an odd number which admits of no other part except that which is called after its own name (n-apiiJniftov <avri2)." The prime numbers art; 3, 5, 7 etc., and (here is no submuUiple of 3 except ^rd, no subtnultipiL' of 1 1 except y j th, and so on. hi all these cases the only submultiple is an unit. According to Nicomachus 3 is the first prime number, whereas Aristotle {Topics viii. 2, 157 a 39) regards a as a prime number ; "as the dyad is the only even number which is prime," showing that this divergence from the l^ythagortan doctrine was earlier than Euclid. The number 2 also satisfies Euchd's definition of a prime number, lamblichus (p. 30, 27 sqq.) makes this the ground of another attack upon Euclid. His argument (the text of which, however, leavijs much to be desired) appears to be that i is the only even number which has no other part except an unit, while the subdivisions of the even, as previously explained by him (the ti-en-timts even, the even-odd, and odd-even), all exclude primeness, and he has previously explained that 2 is pottniially even^xld, being obtained by multiplying by 2 i\\^ potentiaiiy odd, i.e. the unit; hence 2 is regarded by him as bound up with the subdivisions of even, which exclude primeness. 'I'heon seems to hold the same view as regards i, but supports it by an ap|)arent circle. A prime number, he says (p. 23, 14 — 23), is also called odd-times odd; therefore only odd numbers are prime and in composite. Even numbers are not measured by the unit alone, except i, wliich therefore (p. 24, 7) is odd-Aiti' {itiparirati&rfi) without being prime. A variety of other names were applied to prime numbers. We have already noted the curious designation of them as add-titnts odd. According to lamblichus (p. 27, 3 — 5) some called them evthymtiric ((uSu^fTpwds), and Thymaridas rectilinear (<iSv7pa/tfuitOT), the ground being that they can only be set out in one dimension with no breadth (iirXar^s yap iv i-g Mtati iiji' iv liovot iiuTTafitun). The same aspect of a prime number is also expressed by Aristotle, who (Metaph. 1020 b 3) contrasts the composite number with that which is only in one dimension (fioi'oi' i<^ tv cuv). Theon of Smyrna (p. 23, 1 2) gives ■)>pa^/iutdt {linear) as the alternative name instead of cv^uypo^oidt. In either ease, to make the word a proper description of a prime nun>ber we have to understand the word only ; a prime number is that which is linear, or rectilinear, only. For Nicomachus, who uses the form linear, expressly says (11. 13, 6) that all numbers are so, i.e. all can be represented as linear by dots to the required amount placed in a line. A prime number was called prime or first, according to Nicomachus (1. II, 3), because it can only be arrived at by putting together a certain number of units, and the unit is the beginning of number (cf. Aristotle's second sense of irpiuTos "as not being composed of numbers" wi /jltj o-uymurSat i( dpiSfiMv, Anal. Post. \\. 13, 96 a 37), and also, according to lamblichus, because there is no number before it, being a collection of units (^ofaSur <riim]yut), of which it is a multiple, and it appears firsl as a basis for other numbers to be multiples of. Definition 12. EtpioT'Ol Jrpot a\AijXou! apSpai vaw oi /iom& judv]) fttTpoi,! - ■jt koh-iJ liirpi^. By way of further emphasising the distinction between ''prime" and "prime to one another," Theon of Smyrna (p. 23, 6—8) calls the former " prime aholtttefy " (dwKm), and the latter " prime to one another and not 016 - . BOOK Vn ■ .. [V!I. DEFP. I!!— 14 absolttfdy" or *^noi in themseives" (oi xaff aarmi). The latter (p. 44, 3 — lo) are " measured by the unit [sc. only] as common measure, even though, taken by themselves {w jrpo? Javrm), they be measured by some other numbere." From Theon's illustrations it is clear that with him as with Euclid a. number prime to another may be even as well as odd. In Nicomachus (i. 1 1, i) and lambllchus (p. id, 19), on the other hand, the number which is " in itself secondary (Scvre^Mt) and composite (tr!ni6tTo%), but in relation to another prime and incomposite," is a subdivision of odd. I shall call more particular attention to this difference of classification when we have reached the definitions of " composite " and " composite to one another " ; for the present it is to be noted that Nicomachus (1. 13, i) defines a number prime to another after the same manner as the absolutely prime ; it is a number which " is measured not only by the unit as the common measure but also by some other measure, and for this reason can also admit of a part or parts called by a difTerent name besides that called hy the same name (as itself), but, when examined in comparison with another number of similar character, is found not to be capable of being measured by a common measure in relation to the other, nor to have the same part, called by the same name as (any of) those simply (air\c«) contained in the other; e.g. 9 in relation to 25, for each of these is in itself secondary and composite, but, in comparison with one another, they have an unit alone as a common measure and no part is called by the same name in both, but the third in one is not in the other, nor is the fifth in the other found in the first." Definition 13. . . SwrftTOt optC/«k ^<rr(v o afiSfu^ tiki /iMrpoufitvot. Euclid's definition of compositt is again the same as Theon's definition of numbers "composite in relation to themselves," which (p. 24, 16) are " numbers measured by any less number," the unit being, as usual, not regarded as a number. Theon proceeds to say that " of composite numbers they call those which are contained by two numbers plane, as being investigated in two dimensions and, as it were, contained by a length and a breadth, while (they call) those (which are contained) by three (numbers) iolid, as having the third dimension added to them," To a similar effect is the remark of Aristotle {Mtlaph. loio b 3) that certain numbers are " composite and are not only in one dimension but such as the plane and the solid (figure) are representations of (^i^iijjiia), these numbers being so many times so many (irocraKK iroa-oi), or so many times so many times so many (iroo-am! wtwaitt* voaai) respectively." These subdivisions of composite numbers are, of course, the subject of Euclid's definitions 17, 18 respectively. Euclid's composite numbers may be either even or odd, like those of Theon, who gives 6 as an instance, 6 being measured by both » and 3. Definition 14. liirpit. Theon (p. 44, 18), like Euclid, defines numbers eomposite to one another as *' those which are measured by any common measure whatever " (excluding unity, as usual). Theon instances 8 and ti, with i as common measure, and 6 and 9, with 3 as common measure. vir. DErF, 14— :6] NOTES ON DEFINITIONS Ji— 16 sS? As hinted above, there is a great difference between Euclid's classification of prime and composite numbers, and of numbers prime and comp>osite to one another, and the classification found in Nicomachus (1. 11 — 13) and lamblichus. According to the latter, all these kinds of numbers are sub- divisions of the class of odd numbers only. As the class of even numbers is divided into three kinds, (i) the even-times even, (2) the even-odd, which form the extremes, and (3) the odd-even, which is, as it were, intermediate to the other two, so the class of odd numbers is divided into thtee, of which the third is again a mean between two extremes. The three are : (i) t\\t primt and ineomposite, which is like Euclid's prime number except that it excludes 2 ; {i) the ieconiary and composite, which is "odd because it is a distinct part of one and the same genus {Sia tu ii jvot kcu tov avroii yirous Stciit<Kpur0(u) but has in it nothing of the nature of a first principle (ap;(«iSft) ; for it arises from adding some other number (to itself), so that, besides having a part called by the same name as itself, it possesses a part or parts called by another name." Nicomachus cites 9, 15, 21, 25, 27, 33, 35, 39. It is made clear that not only must the factors be both odd, but they must all be prime numbers. This is obviously a very inconvenient restriction of the use of the word composite, a word of general signification. (3) is that which is "secondary and composite in itself but prime and ineomposite to another" The actual words in which this is defined have been given above in the note on Def. 12. Here again all the factors must be odd and prime. Besides the inconvenience of restricting the term composite to odd numbers which are composite, there is in this classification the further serious defect, pointed out by Nesselmann {Die Algebra der Griechen, 1842, p. 194), that subdivisions (2} and (3) overlap, subdivision (2) including the whole of subdivision {3). The origin of this confusion is no doubt to be found in Nicomachus' perverse anxiety to be symmetrical ; by hook or by crtxik he must divide odd numbers into three kinds as he had divided the even. lamblichus (p. 28, 13) carries his desire to be Ic^cal so far as to point out why there cannot be a fourth kind of number contrary in character to (3), r\amely a number which should be "prime and ineomposite in itself, but secondary and composite to another " ! Definition 15. 'ApiB/iAi dpSfthv iroXAaTrAxuruL^cif Xiynat, oraf, 3crot »uriv h earr^ )tov^K, T«ravrax(! inivTi$^ o iraXXaiTAairiafD^ci'Ov, ««! yiyijrrai tk. This is the well known primary definition of multiplication as an abbreviation of addition. Definition 16. iviirtoo^ KaktiTat^ irAcvpat Bi airroZ ol vo\kair\aeTaurayT€^ dXXijKom dpiBfioL The words plane and solid applied to numbers are of course adapted from Iheir use with reference to geometrical figures. A number is therefore called linear (ypaiifuKVi) when it is regarded as in one dimension, as being a lengtli a«8 BOOK vn [vii. def. i6 (lAiJKiys), When it takes another dimension in addition, namely breadth (TrAaros), it is in two dimensions and becomes plane (in-iVtScn). The distinction bet wet! n a plane and a plane number is marked by the use of the neuter in the former case, and the masculine, agreeing with aptS/io!, in the latter case. So witli a square and a square number, and so on. Tlie most obvious form of a plane number is clearly that corresponding to a rectangle in geometry ; the number is the product of two linear numbers regarded as sides {TrKfvpai) forming the length and breadth respectively. Such a number is, as Aristotle says, "so many times so many," and a plane is its counterpart {/ii/iTj/ia). So I'lato, in the Thcaeieiiis (147 E — 148 b), says : "We divided all numbers into two kinds, ( i ) that which can be expressed as equal multiplied by equal (tov ivvi.\i.(vav Xaov Xaixtx yiyno-Sai), and which, likening its form to the square, we called square and equilateral ; {«) that which is intermediate, and includes 3 and 5 and every number which cannot be expressed as equal multiplied by e<|uat, but is cither less times more or more times less, being always "contained by a greater and a less side, which number we likened to the oblong figure (ttpo/ijjkh axnt^'") and calleo an obhng number.... Such lints therefore as square tlie equilateral and plane nujnber fi.e, which can form a plane number with equal sides, or a square] we defined as length iltijxo'i) ; but such as square the oblong (here fTfpo/iijKJ)^) [i.e, the square of which is equal to the oblong] we called roots {^vulitai) as not being com- mensurable with the others in length, but only in the plane areas (^irwiBow), to which the squares on them are equal {a Suyavrai)." This passage seems to make it clear that Plato would have represented numbers as Euclid does, by straight lines proportional in length to the numbers they represent (so far as practicable) ; for, since 3 and S are with Plato oblong numbers, and lines with him represent the sides of oblong numbers (since a line represents the " root," the square on which is equal to the oblong), it follows that the unit representing the smaller side must have been represented as a line, and 3, the larger side, as a line of three times the length. But there is another possible way of representing numbers, not by lines of a certain length, but hypain/s disposed in various ways, in straight lines or otherwise. lamblichus tells us (p. 56, 27) that " in old days they represented the ijuantuplicities of number in a more natural way (^ucriKuir (;»>') by splitting them up into units, and not, as in our day, by symbols" (o-«ftj3oXuiiu?). Aristotle too (Metaph. 1092 b 10) mentions one Eurytus as having settled what number belonged to what, such a number to a man, such a number to a horse, and so on, "copying their shapes" (reading rovruii', with Zeller) ^' with pebbles {rms ^^mi), just as those da who arrange numbers in the forms of triangles or squares." We accordingly find numbers represented in Nicomachus and Theon of Smyrna by a number of a's ranged like points according to geometrical figures. According to this system, any number could be represented by points in a straight line, in which case, says lamblichus {p. 56, 26), we shall call it rectilinear because it is without breadth and only advances in length (oirXaTttfi inl isavav to ^7k« vpotunv). The prime number was called by Thymaridas rectilinear par excellence, because it was without breadth and in one dimension only (iift fc ^vov SiuTTOfitvin). By this must hi meant the impossibility of representing, say, 3 as a plane number, in Plato's sense, i.e. as a product of two numbers corresponding to a rectangle in geometry ; and this view would appear to rest simply upon the representation of a number by points, as (iistinct from lines. Three dots in a straight line would have no breadth ; and if breadth were introduced in the sense of producing a rectangle, i.e. by placing the same VII. DEF. i6] NOTE ON DEFINITION 16 "^ number of dots in a second line below the first line, the first f/ane number would be 4, and 3 would not be a plane number at ali, as Plato says it is. It seems therefore to have been the alternative representation of a number by points, and not lines, which gave rise to the different view of a plane number which we find tn Nicomachus and the rest. By means of separate points we can represent numbers in geometrical forms other than rectangles and squares. One dot with two others symmetrically arranged below it shows a triangle, which is a figure in two dimensions as much as a rectangle or parallelogram is. Similarly we can arrange certain numbers in the form of regular ptnlagons or other polygons. According therefore to this mode of representation, 3 is the first plane number, being a triangular number. The method of formation of triangular, square, pentagonal and other polygonal numbers is minutely described in Nicomachus (11. 8 — 11), who distinguishes the separate series of gnomons belonging to each, i.e. gives the law determining the number which has to be added to a polygonal number with n in a side, in order to make it into a number of the same form but with n + i in a side (the addend being of course the gnomon). Thus the gnomon ic series for triangular numbers is '> 2i 3) At 5"- 't that for squares i, 3, 5, 7... ; that for pentagonal numbers I, 4, 7, 10,,., and so on. The subject need not detain us longer here, as we ate at present only concerned with the different views of what constitutes a plane number. Of plane numbers in the Platonic and Euclidean sense we have seen that Plato recognises two kinds, the square and the oblong (vpoii-^icf}<s or htpoinjieifs). Here again Euclid's successors, at all events, subdivided the class more elaborately. Nicomachus, Theon of Smyrna, and lamblichus divide plane numbers with unequal sides into (i) Irtpofi-iJKtK, the nearest thing to squares, viz. numbers in which the greater side exceeds the less side by i only, or numbers of the form n(n+ i), e.g. i . J, a - 3, 3 . 4, etc. {according to Nico- machus), and (2) wpo{tT^Kfi<s, or those whose sides differ by z or more, i.e. are of the form n(n + m), where m is not less than z (Nicomachus illustrates by 2 . 4, 3 . 6, etc.). Theon of Smyrna (p. 30, 8 — 14) makes wpofujuii! include Irf/jo/iijMtt, saying that their sides may differ by i or more; he also speaks of parallelogram- numbers as those which have one side different from the other by 3 or more ; I do not find this latter term in Nicomachus or lamblichus, and indeed it seems sufterfluous, as parallelogram is here only another name for oblong, lamblichus (p. 74, z 3 sqq,), always critical of Euclid, attacks him again here for confusing the subject by supposing that the htpofi^inp number is the pro- duct of any two different numbers multiplied together, and by not distinguishing the oblong (irpo^ifiti;;) from it : " for his definition declares the same number to be square and also htpoit^xtft, as for example 36, 16 and many others : which would be equivalent to the odd number being the same thing as the even." No importance need be attached to this exaggerated statement ; it is in any case merely a matter of words, and it is curious that Euclid does not in feet use the word fr«po/iijir7t of numiers at all, but only of geometrical oblong figures as opposed to squares, so that lamblichus can apparendy only have inferred that he used it in an unorthodox manner from the geometrical use of the term in the definitions of Book i. and from (he fact that he does not give the two subdivisions of plane numbers which are not square, but seems only to divide plane numiers into square and not-square. The aigument that Ircpo^ijKttf numbers are a natural^ and therefore essential, subdivision lamblichus appears to fotmd on the method of successive addition by which they can be evolved ; as square numbers are obtained by successively adding i^o •'-■ BOOK VII [vii. DEFF. i6, 17 odd numbers as gnomons, so ir<pci/ij)Kf« are obtained by adding even numbers as gnomons. Thus i.z = 2, 2.3 = 2 + 4, 3,4-2 + 4 + 6, and so on. Definition 17. {TTfpm i(mv, v\tvpal Sc airoO oi jroAAairXacricio-aiTf! dAAi^Aou; apSfioi. What has been said of the two apparently different ways of regarding a plane number seerns to apply equally, mutatis mutandis, to the definitions of a solid number. Aristotle regards it as a number which is so many times so many times so many (jrpcrojtu irt«rai«! itoaoi). Plato finishes the passage about lines which represent the sides of square numbers and lines which are roots i^vva/itK), i.e. the squares on which are equal to the rectangle representing a number which is oblong and not square, by adding the words, " And another similar property belongs to solids " (kqI ircpi to o-rtpta aAAo toioZtov). That is, apparently, there would be a corresponding term to root (ftJca/in) — practically representing a surd— to denote the side of a cube equal to a parallelepiped representing a solid number which is the product of three factors but not a cube. Such is a solid number when numbers are represented by straight lines : it corresponds in general to a parallelepiped and, when all the factors are equal, to a cube. But again, if numbers be represented by points, we may have solid numbers (i.e. numbers in three dimensions) in the form of pyramids as well. The first number of this kind is 4, since we may have three points fonTiing an equilateral triangle in one plane and a fourth point placed in another plane. The length of the sides can be increased by i successively ; and we can have a series of pyramidal numbers, with triangles, squares or polygons as bases, made up of layers of triangles, squares or similar polygons respectively, each of which layers has one less in the side than the layer below it, until the top of the pyramid is reached, which of course, is one point representing unity, Nicomachus (11, 13 — 16), Theon of Smyrna {p. 41 — 2), and lamblichus (P- 9S> '5 sqq.), all give the different kinds c>{ pyramidal solid numbers in addition to the other kinds. These three writers make the following further distinctions between solid numbers which are the product of three factors. 1. First there is the equal by equal by equal (Icrant to-aittt urof), which is, of course, the cube. 2. The other extreme is the unequal by unequal by unequal {aVto-ciKis o.vuia.Ki.% ai'io-ot), or that in which all the dimensions are different, e.g. the product of 2, 3, 4 or 2, 4, 8 or 3, 5, la. These were, according to Nicomachus (11, 16), called scalene, while some called them a^tjvlanoi (wedge-shaped), others tr^i/KUTKot (from tr^ij'f, a wasp), and others ^mitlantot {altar-shaped). Theon appears to use the last term only, while lamblichus of course gives all three names. 3. Intermediate to these, as it were, come the numbers " whose planes form frepo/nJ«(i numbers" (i.e. numbers of the form*«(« + i)). These, says Nicomachus, are QaXXei parallelepipedal. Lastly come two classes of such numbers each of which has two equal dimensions but not more. VII. DEPP. 17— 19I NOTES ON DEFINITIONS id— 19 i0 4. If the third dimension is less than the others, the number b efual ly tquai iy less (uraKtt itroi i'Aairoi'aicw) and is called a plinth (nrXicSw), e.g. 8.8.3. 5- If the third dinnensioti is greatei than the others, the number is equal by equal by greater (ktb'ms ujik fniioraitw) and is called a beam (&okk), e.g, 3.3.7. Another name for this latter kind of number (according to lambltchus) was ernjXU (diminutive of onjXij). Lastly, in connexion with pyramidal numbers, Nifcomachus (11. 14, 5) dis- tinguishes numbers corresponding X^i frusta of pyramids. These are truncated (mi^upoi), twUe-truntated (BtKoXoupot), thrict-fruncated (rpiKoAoupm) pyramids, and so on, the term being used mostly in theoretic treatises («v mr/ypdiiftairi fiakuTTa To« StiofnifumKoU). The truncated pyramid was formed by cutting off the point forming the vertex. The twice-truncaied was that which lacked the vertex and the next plane, and so on. Theon of Smyrna (p. 42, 4) only mentions the truncated pyramid as " that with its vertex cut off" (ij ttjc Kofsii^ijv diroTiTixsjiLivrj), saying that some also called it a trapezium, after the similitude of a plane trapezium formed by cutting the top off a triangle by a straight line parallel to the base. Definition 18. Tcrpaywvo; &.pSlx6% i<rr\y h (cranf uro( ^ [6j vtro Svo wrtav &pi$ftiZv ntpi- A particular kind of square distinguished by Nicomachus and the rest was the square number which ended (in the decimal notattor) with the same number as its side, e.g. i, 25, 36, which are the squares of r, 5 and 6. These square numbers were called cyclic (kukXmw) on the analogy of circles in geometry which return again to the point from which they started. Definition 19, KiJjSps Si 6 ttraKit ta-ov liraicK ^ [i] inro rpiuv t(Fii>r Jpifl/uuv wipiixontum. Similarly cube numbers which ended with the same number as their sides, and the squares of thosr sides aisa, were called spherical (o-f^tpiKof) or reatrrtnt {a.iraiaaassta.fma\). One might have expected that the term spherical would be applicable also to the cubes of numbers which ended with the same digit as the side but not necessarily with the same digit as the square of the side also. E.g, the cube of 4, i.e. 64, ends with the same digit as 4, but not with the same digit as 1 6, But apparently 64 was not called a spherical number, the only instances given by Nicomachus and the rest being those cubed from numbers ending with 5 or 6, which end with the same digit if squared. A spherical number is in fact derived from a circular number only, and that by adding another equal dimension. Obviously, as Nesselmann says, the names cyclic and spherical applied to numbers appeal to an entirely different principle from that on which the figured numbers so far dealt with were formed. jtga BOOK VII 1 1 /• . [vii. DEF. ao Definition 2a £(7ttic($ ]j iroXAairXdmof i^ to awo fiipo^ ^ ra aura ftc^ ONrtv, Euclid does not give in this Book any definition of latio, doubtless because it could only be the same as that given at the beginning of Book v., with numbers substituted for "homogeneous magnitudes " and "in respect of size" {njXiKonp-a) omitted or altered. We do not find that Nicomachus and the rest give any substantially diflerent definition of a ra/io between numbers. Theon of Smyrna says, in fact (p, 73, 16), that " ratio in the sense of proportion (K6yoi 4 itar' ara'Xoyok) is a sort of relation of two homogeneous terms to one another, as for example, double, triple." Similarly Nicomachus says {11. II, 3) that "a ratio is a relation of two terms to one another," the word for " relation " being in both cases the same as Euclid's {cr)(i<nt}. Theon of Smyrna goes on to classify ratios as greater, less, or equal, i.e. as ratios of greater inequality, less inequality, or equality, and then to specify certain arithmetical ratios which had special names, for which he quotes the authority of Adrastus. The names were iroXXaTrXoo'tof, irtfioptoif ivifitp^, iroAAairAofrtcn^topiof} iroXXar\iiurt*rtfAM(«j<! (the first of which is, of course, a multiple, while the rest are the equivalent of certain types of improper fractions as we should call them), and the reciprocals of each of these described by prefixing vini or fui. After describing these particular classes of arithmetical ratios, Theon goes on to say that numbers still have ratios to one another even if they are different from all those previously described. We need not therefore concern ourselves with the various types ; it is sufficient to observe that any ratio between numbers can be expressed in the manner indicated in Euclid's definition of arithmetical proportion, for the greater is, in relation to the less, either one or a combination of more than one of the three things, (i) a multiple, {3} a submultiple, (3) a proper fraction. It is when we come to the definition of proportion that we begin to find differences between Euclid, Nicomach us, Theon and lamblichus, " Proportion," says Theon (p. 81, 6), " is similarity or sameness of more ratios than one," which is of course unobjectionable if it is previously understood what a ratio is ; but confusion was brought in by those {like I'hiasyllus) who said that there were tkret proporiitms (aVoXo^iai), the arithmetic, geometric, and harmonic, where of course the reference is to arithmetic, geometric and harmonic means (litcronfut). Hence it was necessary to explain, as Adrastus did (Theon, p. 106, 15), that of the several mtatts "the geometric was called both proportion /ar extelience and primary... though the other means were also commonly called proportions by some writers." Accordingly we have Nicomachus trying to extend the term " proportion " to cover the various meam as well as a proportion in three or four terms in the ordinary sense. He says (it. »i, 3): " Proportion, /ar «ciy/An« (kv/jhus), is the bringing together (<ni>VXt)^i;) to the same (point) of two or more ratios \ or, more generally, (the bringing tc^ether) of two or more relations {tr^ifn^, even though they be subjected not to the same ratio but to a difference or some other (law)." lamblichus keeps the senses of the word more distinct. He says, like Theon, that " proportion is similarity or sameness of several ratios " (p. 98, 14), and that " it is to be premised that it was the geometrical (proportion) which the ancients called proportion par excelknct^ though it is now common to apply the name genemlly to all the remaining means as well " (p. 100, 15). Pappus vit. DEFF. 20— ij] NOTES ON DEFINITIONS 70—21 393 remarks {in. p. 70, 17), "A mean differs from a proportion in this respect tha^ if anything is a proportion, it is also a mean, but not conversely. For there are three means, of which one is arithmetic, one geometric and one harmonic." The last remark implies plainly enough that there is only one proportion (d™Aoy«i) in the proper sense. So, too, says lamblichus in another place (p. 104, 19): "the second, the geometric, mean has been called proportion par excclletue because the terms contain the same ratio, being separated according to the same proportion (aVcl tov airov Xoyov Swaron-t?)." The natural conclusion is that of Nesselmann, that originally the geometric proportion was called ivakoyia, the others, the arithmetic, the harmonic, etc, tneans ; but later usage had obliterated the distinction. Of proportions in the ancient and Euclidean sense Theon fp, 82, 10) distinguished the continuous (trui-tjf^!) and the separated (Zijjpijfi.iv'ri), using the same terms as Aristotle {Eth. I^t'c. 1131 a 32). The meaning is of course clear : in the continuous proportion the consequent of one ratio is the ante- cedent of the next ; in the separated proportion this is not so. Nicomachus (11. 21, 5 — 6) uses the words (onnected (<rwrijLit.ivi}) and disjoined (Snitvyfiivi)) respectively. Euclid r^ularly speaks of numbers in continuous proportion as " proportional in order, or successively " {ifij« liraXoyoi'). Definition 21. 'QfioiOi ^lirtBoi KoX <rr€pto\ afii$ftOL ihnv al drvdXoyov c^orrc; ras rXcupa^. Theon of Smyrna remarks (p, 36, 12) that, among plane numbers, at/ squares are similar, while of lTtpofi-^iif!.<! those are similar " whose sides, that is, the numbers containing them, are proportional." Here irfpofirfKiji must evidently be used, not in the sense of a number of the form n{n + i), but, as synonymous with irpofHjieij;, any oblong number ; so that on this occasion Theon follows the terminology of Plato and (according to lamblichus) of Euclid. Obviously, if the strict sense of rrfpo/ufitiji is adhered to, no two numbers of that form can be similar unless they are also c^uat. We may compare lamblichus' elaborate contrast of the square and the irfpo/tijuj^. Since the two sides of the square are equal, a square number might, as he says {p. 8a, 9), be fitly called ISio^iJxj;! (Nicomachus uses rauro^jjiojt) in contrast to iripoiiTJinji ; and the ancients, according to him, called square numbers " the same " and " similar " (rajirou; re nai ofiotavt), but iTtpo/ii^KtK numbers " dissimilar and other " (ovo^oi'ous koI Baripov^), With regard to solid numbers, Theon remarks in like manner (p. 37, 2) that atl cube numbers are similar, while of the others those are similar whose sides are profwrtional,- i.e. in which, as length is to length, so is breadth to breadth and height to height. DEFmiTION 22. Theon of Smyrna (p. 45, 9 sqq.) and Nicomachus (i. 16) both give the same definition of a perfect number, as well as the law of formation of such numbers which Euclid proves in the later proposition, ix. 36. They add however definitions of two other kinds of numbers in contrast with it, (i) the oTier-pnfect (wVfpT^Aijv in Nicomachus, virtprtXttot in Theon), the m - .. BOOK VII . ;,,...,, •r* ;-I Ir-.V sum of whose parts, i.e. submultiples, is greater than the numbttr itself, e.g. 1 1, 24 etc., the sum of the parts of 12 being 6+4 + 3 + 2+1 = 16, and the sum of the parts of 34 being 12 + 8 + 6 + 4 + 3 + 2 + i = 36, (2) the defective (AAiTTiTv), the sum of whose parts is less than the whole, e.g. S or 14, the parts in the first case adding up to 4 + a + i, or 7, and in the second case to 7 + 2 + I, or 10. All three classes are however made by Theon subdivisions of numbers in general, but by Nicouiachus subdivisions of even numbers. The term perfect was used by the Pythagoreans, but in another sense, of 10; while Theon tells us (p. 46, 14) that 3 was also called perfect "because it is the first number that has beginning, middle and extremity; it is also both a line and a plcme (for it is an equilateral triangle having each side made up of two units), and it is the first Imk and potentiality of the solid (for a solid must be conceived of in three dimensions)." There are certain unexpressed axioms used in Book vii. as there are in earlier Books. The following may be noted, I. If j^ measures B, and ^ measures C, A will measure C. a. \l A measures B, apd also measures C, A will measure the difference between B and C when they are unequal. 3, If A measures B, and also measures C, A will measure the sum of B and C. It is clear, from what we know of the Pythagorean theory of numbers, of musical intervals expressed by numbers, of difTerent kinds of means etc., that the substance of Euclid Books vii.— ix. was no new thing but goes back, at least, to the Pythagoreans. It is well known that the mathematics of Plato's Jlmaeus is essentially Pythagorean. It is therefore a priori probable (if not perhaps quite certain) that Plato irvSayopi'iet even in the passage (32 a, a) where he speaks of numbers " whether solid or square " in continued proportion, and proceeds to say that between planes one mean suffices, but to connect two solids two means are necessary. This passage has been much discussed, but I think that by " planes " and " solids " Plato certainly meant square and iolid ntim&ers respectively, so that the allusion must be to the theorems estabtished in Eucl. viii. ti, 12, that between two square numbers there is one mean proportional number, and between two cube numbers there are two mean proportional numbers'. ..-1. . . Iv ' It is true that similar p!ine and solid numbers have the $arne property (Eucl, viii. 18, 19) ; but, if Plato had meant similar pkne and solid numbers generally, I think il would have been necessary to specify that they were " similir," whereas, seeing that the Timams a as a whole concerned with regular fi|,nires, there is nothing unnatural in allowing rc^lar or equilaleral to be understood. Further Plato speaks first of Juni/ini and iytoi and then of "planes" [tTrlriia) and "solids" (Fttpti.) in such a way as to surest that ixiviiua cor- respoiid to iwlTttSu, and 6n/Kot to vripti. Now the regular meaning m J^a^ii is square (or sometimes square rant), and I think it is here used m the sense of sauarr, notwithstanding that Plato seems to speak of lAm squares in continued proportion, whereas, in general, the mean between two squares as eitremes would not be square but olJong. And, if Suti/uu are squares, it is reasonable to suppose that the tyKot afe also ei/iiiiateral, i.e. the "sulids" are cubes. 1 am aware that Tb. Habler (Biilisthtia Malkimatita, VIII3, 1008, pp. 173—4) thinks that [he passage is to be explained by reference to the problem of the duplication of the cube, and does not refer to numbers at all. Against this we have to put the evidence of Nicomachus (It. 54, 6) who, in speaking of "a certain Platonic theorem," quotes the very same results of Eud. VIii. 11, n. Secondly, it is worth noting that Hiiblet's explanation is dulinctly raled out by Democritus tb« Platonjst (jrd cettt. A,D.} who, according to Proclus HISTORICAL NOTE i^S It is no less clear that, in his method and line of argument, Euclid was following earlier models, though no doubt making improvements in the ex- position. The tract on the &ciio Cnn&nis, KOTaro/ii; KavoviK (as to the genuine- ness of which see above, Vol. !., p. 17) is in style and in the form of the propositions generally akin to the Ekmenis. In one proposition (2) the author says "ife learned ((ftaSo^ei-) that, if as many numbers as we please be in (con- tinued) proportion, and the first measures the last, the first will also measure the intermediate numbers " ; here he practically quotes EUm. viii. 7- In the 3rd proposition he proves that no number can be a mean between two numbers in the ratio known as hrmopitu, the ratio, that is, of » ■(■ i to n, where « is any integer greater than unity. Now, fortunately, Boethius, De vistitufione viuska. III. 1 1 {pp. 885—6, ed. Friedlein), has preserved a proof by Archytas of this same proposition ; and the proof is substantially identical with that of Euclid. The two proofs are placed side by side in an article by Tannery {Sibliothcta Mathematiea, vr,, 1905/6, p. 227). Archytas writes the smaller term of the proportion first (instead of the greater, as Euclid does). Let, he says, ^, ^ be the " superparticularis proportio " (iu-ifiopio!' hasmfii-a in Euclid). Take C, £>£ the smallest numbers which are in the ratio of A to B. [Here DE means D + E: and in this respect the notation is different from that of Euclid who, as usual, takes a line DF divided into two parts at G, GF corresponding to E, and DG to D, in Archytas' notation. The step of taking C, DE, the smallest numbers in the ratio of A to B, presupposes Eucl. vii. 33 J Then DE exceeds C by an aliquot part of itself and of C [cf the definition of iTcifiopuK dpi^/ia^ in Nicomachus, i. 19, i]. Let D be the excess [i.e. E is supposed equal to C]. " I say that D is not a number but an uniL" For, if Z" is a number and a part of DE, it measures JJE ; hence it measures E, that is, C- Thus JD measures both C and BE, which is impossible ; for the smallest numbers which are in the same ratio as any numbers are prime to one another. [This presupposes Eucl, vei. 2a.] There- fore -D is an unit ; that is, DE exceeds C by an unit. Hence no number can be found which is a mean between two numbers C, DE. Therefore neither can any number be a mean between the original numbers A, B which are in the same ratio [this implies Eucl. vii. 20]. We have then here a clear indication of the existence at least as early as the date of Archytas (about 430 — 365 B.C.) of an Eitments of Aritkmetie in the form which we call Euclidean ; and no doubt text-books of the sort existed even before Archytas, which probably Archytas himself and Eudoxus improved and developed in their turn. {In P!at<snis Tiaamm (ommtntaria, [+9 c), said that the dlfficultiep of the passage of the TitHtuui tiad misUd some people into connecting it with tbe duplication of the cube, whereas it really referred to similar planes and solids with sides in ra/iotial numbers^ Thirdly, I do not think that, under tht supposition that the Delian problem is referred to, we get the required sense. The problem in that case is not that of finding two mean proportionals Between two eudii Ijut that of finding a second cybe the content of which ahall Ue equal to twice, or k times (where ^ is any numtier not & complete cube), the content of a given Oi\x (^. Two mean proportionals are found, not between cubes, but between two siraight linn in the ratio of 1 to k, or between a and ks. Unless .( is a culje, there would lie no point in saying that two means are necessary to connect t aad k, and not one mean ; for ijk is no more naluial than .Ji, and would be less natural in the case where * happened to t>e square. On the other hand, if ^ is a cube, ^ that it Is a question of finding means between tuhe numbers, the dictum of Plato is perfectly intelligible ; nor is any real difficulty caused by the generality of the statement that two means are al-ivays necessary to connect them, because any property enunciated generally of two cut>e numbers should obviously be true of cubes 9S sikH, that is, it must hold in the extreme ease of two cubes which are ^me to aw att&ihir. BOOK VII. PROPOSITIONS. A H F ■ O H Proposition i. Two UHsqtiai numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the original numbers mill be prime to one another. For, the less of two unequal numbers AB, CD being continually subtracted from the greater, let the number which is left never measure the one before it until an unit is left ; I say that AB, CD are prime to one another, that is, that an unit alone measures AB, CD. For, liAB, CD are not prime to one another, some number will measure them. Let a number measure them, and let it be E\ let CD, measuring BF, leave FA less than itself, let AF, measuring DG, leave GC less than Itself, and let GC, measuring FH, leave an unit HA. Since, then, E measures CD, and CD measures BF, therefore E also measures BF. But it also measures the whole BA ; therefore it will also measure the remainder ..<^^. But AF measures DG ; therefore E also measures DG. VII. i] PROPOSITION I *j$)l But it also measures the whole DC ■ therefore it will also measure the remainder CG. But CG measures FH ; therefore E also measures FH. But it also measures the whole FA ; therefore it will also measure the remainder, the unit AH, though it is a number : which is impossible. Therefore no number will measure the numbers ^5, CD; therefore AB, CD are prime to one another. [vn. Def. i a] Q. E, D. It is proper to remark here that the representation in Books vn. to ix. of numbers by straight lines is adopted by Heiberg from the mss. The method of those editors who substitute poirtti for lines is open to objection because it practically necessitates, in many cases, the use of specific numbers, which is contrary to Euclid's manner. "Let CD, measuring BF, leave FA less than itself." This is a neat abbreviation for saying, measure along BA successive lensths equal to CD until a point F is reached such that the length FA remaining is less than CD ; in other words, let BF be the largest eitact multiple of CD corstained in BA. Euclid's method in this proposition is an application to the particular case of prime numbers of the method of finding the greatest common measure of two numbers not prime (t) one another, which we shall find in the next proposition. With our notation, the method may be shown thus. Supposing the two numbers to be a, b, we have, say, t ' If now a, 3 are not prime to one another, they must have a commcm measure t, where e is some integer, not unity. And since t measures a, 6, it measures a ~pb, i.e. (, Ag^n, since t measures b, c, it measures 6-fc, i.e. d, and lastly, since * measures <■, d, it measures e~rd, i.e. i; ' i . 1. if which is impossible. Therefore there is no integer, except unity, that measures a, b, which are accordingly prime to one another. Observe that Euclid assumes as an axiom that, if a, b are both divisible by f, so is a -pb. In the next proposition he assumes as an axiom that c will in the case supposed divide a +pb, c ■F »98 BOOK Vn [vu. 1 Proposition 2. Given (wo numbers not prime to one another, to find their greatest contnion measure. Let AB, CD b« the two given numbers not prime to one another. Thus it is required to find the greatest J^ common measure of AB, CD. If now CD measures AH — and it also ^ measures itself — CD is a common measure of CD, AB. And it is manifest that it is also the greatest ; for ao greater number than CD will measure ^ ^ CD. But, if CD does not measure AB, then, the less of the numbers AB, CD being continually subtracted from the greater, some number will be left which will measure the one before it. For an unit will not be left ; otherwise AB, CD will be prime to one another [vii. i], which is contrary to the hypothesis. Therefore some number will be left which will measure the one before it. Now let CD, measuring BE, leave EA less than itself, let EA, measuring DF, leave EC less than itself, and let CA" measure AE. Since then, CF measures AE, and AE measures DF^ therefore CF will also measure DF. But it also measures itself; therefore it will also measure the whole CD. But CD measures BE ; therefore C/^also measures BE. But it also measures EA ; therefore it will also measure the whole BA. But it also measures CD ; therefore CF measures AB, CD. Therefore CF'is a common measure oi AB, CD, VII. 2] PROPOSITION 2 -4$9 I say next that it is also the greatest. For, if CF is not the greatest common measure of AB, CD, some number which is greater than CF will measure the numbers AB, CD. Let such a number measure them, and let it be G. Now, since G measures CD, while CD measures BE, G also measures BE. "'" But it als© measures the whole BA ; therefore it will also measure the remainder AB, But .^^ measures /?y^; therefore G will also measure DF. But it also measures the whole DC ; therefore it will also measure the remainder CF, that is, the greater will measure the less : which is impossible. Therefore no number which is greater than Ci^ will measure the numbers AB, CD ; therefore CF is the greatest common measure of AB, CD, PORISM. From this it is manifest that, if a number measure two numbers, it will also measure their greatest common measure. Q. E. D.. Here we have the exact niethod of finding the greatest common measure given in the text- books of algebra, including the reductio ad abmrdum proof that the number arrived at is not only a common measure but the greatest common measure. The process of finding the greatest common measure is simply shown thus : P± €)biq •■ -. We shall arrive, says Euclid, at some number, say d, which measures the one before it, i.e. such that e = rd. Otherwise the process would go on until we arrived at unity. This is impossible because in that case a, b would be prime to one another, which is contrary to the hypothesis. Next, like the text-books of algebra, he goes on to show that d will be some common measure of a, b. For d measures c ; therefore it measures jff + i^, that is, #, ■ ' and hence it measures pb + ^, that is, a. Lastly, he proves that d is the greatest common measure of a, b as follows. Suppose that ? is a common measure greater than d. Then e, measuring a, i, must measure a-fb, or c. • Soo BOOK VII [vii. *, 3 Similarly « must measure 6 -qe, that is, d: which is impossible, since e is by hypothesis greater than d. . . i - • . ; Therefore etc. " .-! - . Euclid's proposition is thus identical with the algebrmical proposidon as generally given, e.g. in Todhunter's algebra, except that of course Euclid's numbers are integers. Niconiachus gives the same rule (though without proving it) when he shows how to determine whether two given odd numbers are prime or not prime to one another, and, if they are not prime to one anothet, what is their common measure. We are, he says, to compare the numbers in turn by continually taking the less from the greater as many times as possible, then taking the remainder as many times as jKwsible from the less of the original numbers, and so on ; this process " will finish either at an unit or at some one and the same number," by which it is implied that the division of a greater number by a less is done by separate snbfraetiens of the less. Thus, with regard to 2 1 and 49, Nicomachus says, " I subtract the less from the greater ; a8 is left ; then ^;ain I subtract from this the same 2 r (for this is possible); 7 is left; I subtract this from 2t, 14 is left; from which I again subtract 7 (for this is possible); 7 will be left, but 7 cannot be subtracted from 7." The last phrase is curious, but the meaning of it is obvious enough, as also the meaning of the phrase about ending " at one and the same number." The proof of the Porism is of course contained in that part of the propo- sition which proves that G, a common measure different from CF^ must measure CF. The supposition, thereby proved to be false, that G is greater than CFdxxA not affect the validity of the proof that G measures CF'xn any case. Proposition 3. " ■. Given three numbers noi prime to one another, to find their greatest common measure. Let A, B, C be the three given numbers not prime to one another ; thus it is required to find the greatest common measure oi A, B, C. For let the greatest common measure, D, of the two numbers .^, .5 be taken ; [vu. l] then D either measures, or does not measure, C, First, let it measure iL But it measures A, B also ; therefore D measures A, B, C ; therefore /? is a common measure of .^, B, C, I say that it is also the greatest. E| Fl VII. 3] PROPOSITIONS 2, 3 30I For, if i? is not the greatest commoQ measure of^, B, C, some number which is greater than D will measure the numbers A, B, C. Let such a number measure them, and let it be E. Since then E measures A, B, C, _ , it will also measure A, B ; therefore it will also measure the greatest common measure of A, B. [vii. 2, For.] But the greatest common measure of A, B is D ; therefore E measures D, the greater the less : which is impossible. Therefore no number which is greater than Z? will measure the numbers A, B, C; therefore D is the greatest common measure of A, B, C. Next, let D not measure C ; I say first that C, D are not prime to one another. For, since A, B, C are not prime to one another, some number will measure them. Now that which measures A, B, C will also measure A, B, and will measure D, the greatest common measure oi A, B, [vn. %, Por.] But it measures C also ; ^ therefore some number will measure the numbers D, C\ therefore D, C are not prime to one another. Let then their greatest common measure E be taken. [vii. i\ Then, since E measures D, and /? measures ^, ^, • therefore E also measures A, B. , • But it measures C also ; ' • therefore E measures A, B, C; , therefore .£" is a common measure of A, B, C, I say next that it is also the greatest. For, if E is not the greatest common measure of A, B, C, some number which is greater than E will measure the numbers A, B, C. Let such a number measure them, and let it be F. $oz BOOK VII • [vii. 3 ' Now, since /^measures A, B, C, ■ u * '-i. '• ' it also measures /^, Z? ; ' ' -' ' >.i". therefore it will also measure the greatest common measure of W, B. [vn. 2, Por.] But the greatest common measure of A, B is D ; therefore J^ measures D. And it measures C also ; ' ' '"' . '' therefore /^measures Z>, C; therefore it will also measure the greatest common measure of D, C. [vn. i, Por.] But the greatest common measure of Z?, C is ^ ; therefore /" measures £, the greater the Jess: which is impossible. Therefore no number which is greater than S will measure the numbers A, B, C; therefore £ is the greatest common measure of A, B, C, Q. E. D. Euclid's proof is here longer than we should make it because he distinguishes two cases, the simpler of which is really included in the other- Having taken the greatest common measure, say d, of a, t>, two of the three given numbers a, b, c,h^ distinguishes the cases (i) in which d measures (, (2) in which d does not measure c. In the first case the greatest common measure of d, e'\^ d itself; in the second case it has to be found by a repetition of the process of v 11. 2. In either case the greatest common measure oi a, 6, ( is the greatest common measure of d, c. But, after disposing of the simpler case, Euclid thinks it necessary to prove that, if d does not measure c, d and ( must necessarily have a greatest common measure. This he does by means of the original hypothesis that tf, b, c are not prime to one another. Since they are not prime to one another, they must have a common measure; any common measure of a, * is a measure of d, and therefore any common measure of a, *, r is a common measure of d, c \ hence d, c must have a common measure, and are therefore not prime to one another. The proofs of cases (i) and (2) repeat exactly the same alignment as we saw in vii. 3, and it is proved separately for <f in case (i) and t in case (3), where < is the greatest common measure of d, (, (a) that it is a common measure of a, b, e, (fi) that it is the greaiesi common measure. Heron remarks (an-Nairlzi, ed. Curtze, p. 191) that the method does not only enable us to find the greatest common measure of /Arte numbers ; it can be used to find the greatest common measure of as many numbers Miie VII. 3, 4l PROPOSITIONS 3, 4 303 as we please. This is because any number measuring two numbers also measures their greatest common measure ; and hence we can find the g.c.m. of pairs, then the g.c.m, of pairs of these, and so 00, until only two numbers are lefl and we find the g.c.m. of these. Euchd tacitly assumes this extension in VII. 33, where he takes the greatest common measure ofay many numbers as we phase. Proposition 4. Any number is either a pari or parts of any number, the less of the greater. Let A, BC be two numbers, and let BC be the less ; \ I say that BC is either a part, or parts, of A. For A, BC are either prime to one another or not. First, let A, BC be prime to one another. Then, if BC be divided into the units in it, each unit of those in BC will be some part of A ; so that BC is parts of ^. Next let A, BC not be prime to one another; then .5C either measures, or does not measure, A. If now BC measures A, BC is a part oi A, But, if not, let the greatest common measure D of A, BC be taken ; [vn. 2] and let BC be divided into the numbers equal to D, namely BE, EF, FC. Now, since D measures ^4, /? is a part of A. But D is equal to each of the numbers BE, EF, FC; -i* therefore each of the numbers BE, EF, FC is also a part of A ; so that BC is parts of A. Therefore etc. 1 . 1 The meaning of the enunciation is of course that, if a, b be two numbers of which i is the less, then b is either a submultiple or soirn proper fraction of a. (i) If a, b are prime to one another, divide each into its units ; then b contains b of the same parts of which a contains a. Therefore b is " parts " or i. proper Jradion of a. (i) If a, b be not prime to one another, either b measures a, in which case i is a submuttiple or " part " of a, or, if ^ be the greatest common measure of a, b, we may put a — mg and h=ng, and h will contain « of the same parts (£) of which a contains m, so that b is again "parts," or a. proper fraction, of a. $04 BOOK VII [vii. 5 ■ I , Proposition 5. If a number be a part of a number, and another be the same part of another, the sum will also be the same part of the sum that the one is of the one. For let the number -(4 be a part of BC, and another, D, the same part of another EF that A is of ^C; I say that the sum of A, D '\% also the same part of the sum of BC, EF that A is of BC. For since, whatever part A is of BC, D IS also the same part oi EF, therefore, as many numbers as there are in j5C equal to A, so many numbers are there also in EF equal to D. Let BC be divided into the numbers equal to A, namely BG, GC, and EF into the numbers equal to D, namely EH, HF\ then the multitude of BG, GC will be equal to the multitude of EH, HF. And, since BG is equal to A, and EH to D, therefore BG, EH are also equal to A, D. For the same reason GC, HF 2iX^ also equal to A, D. Therefore, as many numbers as there are in BC equal to A, so many are there also in BC, £"7^ equal to A, D. Therefore, whatever multiple BC \^o\ A, the same multiple also is the sum of BC, EF of the sum of A, D. Therefore, whatever part A is of BC, the same part also is the sum of A, D of the sum of BC, EF. Q. E. D. If a = -6, and e = -d, then it The proposition is of course true for any quantity of pairs of numbers similarly related, as is the next proposition slso ; and both propositions aie used in the extended form in vil 9, 10. A C D Q H B E VII. 6J PROPOSITIONS 5, 6 305 Proposition 6. // a number be parts of a number, and another be the same parts of another, (he sum will also be the same parts of the sum that the one is of the one. For let the number AB be parts of the number C, and another, DE, the same parts of another, F, that AB is of C ; I say that the sum of AB, DE is also the same parts of the sum of C, F that AB is of C For since, whatever parts AB is of C, DE is also the same parts of F, therefore, as many parts of C as there are in AB, so many parts of /^ are there also in DE. Let AB be divided into the parts of C, namely AG, GB, and DE into the parts of ^, namely DH, HE; thus the multitude of AG, GB will be equal to the multitude of Z?jy, HE, And since, whatever part AG is of C, the same part is /JiYof Aalso, therefore, whatever part AG\%o{ C, the same part also is the sum of AG, DH of the sum of C, F. [vii. 5] For the same reason, ^ ' ■ . ; whatever part GB is of C, the same part also js the sum of GB, HE of the sum of C, F, Therefore, whatever parts AB is of C, the same parts also is the sum of AB, DE of the sum of C, F. Q. E. D. If a = — i, ana c = — d, r n n then a + €=- (fi + d). More generally, if OT . Iff V fft f a = ~ b, c= — a, «-- /, n ' ft n then {a +(+ ( + g+ ,,,) = - {i-*-t/+/-i jo6 BOOK VII [vir. 6, ^ In Euclid's proposition m<.n, but tht generality of the result is of course not aiTected. This proposition and the last are complementary to v, i, which proves the corresponding result with multiple substituted for "pari" or Proposition 7, r If a number be that part of a number, which a number subtracted is of a number subtracted, the remainder will also be the same part of the remainder thai the whole is of the whole. For let the number AB be that part of the number CD which AE subtracted is of CF subtracted ; I say that the remainder EB is also the same part of the remainder FD that the whole AB is of the whole CD. For, whatever part AE is of CF, the same part also let EB be of CG. Now siQce, whatever part AE is of CF, the same part also is EB of CG, therefore, whatever part AE is of CF, the same part also is AB of GF. [vii. s] But, whatever part AE is of CF, the same part also, by hypothesis, is AB of CD ; therefore, whatever part AB \% o^ GF, the same part is it of CD also ; ,. . therefore GF is equal to CD. Let CF be subtracted from each ; therefore the remainder GC is equal to the remainder FD. Now since, whatever part AE is of CF, the same part also is EB of GC, while GC is equal to FD, therefore, whatever part AE is of CF, the same part also is ^^of/'Z>. But, whatever part AE is of CF, the same part also is AB oiCD; vii. 7, 8] PROPOSITIONS 6~8 307 therefore also the remainder EB is the same part of the remainder FD that the whole AB is of the whole CD. Q. E. D, If a=^ -li and f=-rf, we are to prove that a~c = -(6-d), a result differing from that of vji. 5 in that minus is substituted for /Ar. Euclid's method is as follows. Suppose that e is taken such that a-^ = -e. (i) Now e=-d. n Therefore a = -{d-¥e), [vii. 5] whence, from the hypothesis, d-ve = b, so that e = b-d, and, substituting this value of^ in (i), we have .i , a-e=^-{b'-d). Proposition 8. If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder will also be the same parts of the remainder that the whole is of the whole. For let the number AB be the same parts of the number CD that AE subtracted is of CF subtracted ; I say that the remainder EB is also the same parts of the re- mainder FD that the whole AB is of the whole CD. For let GH be made equal to AB, Therefore, whatever parts GH is of CD^ the same parts also is AE of CF. Let GHhe, divided into the parts of CD, namely GK, KH, and AE into the parts of CF, namely AL, LE; thus the multitude of GK, KH-^\\\ be equal to the multitude of AL, LE. c f = Q M K N H A L t B 3A8 book vn ' [vii. 8 Now since, whatever part GK is of CD, the same part also is ^Z of C/s '■"-<• while CD is greater than CF, therefore GK is also greater than AL, Let GM be made equal to AL. Therefore, whatever part GK is of CD, the same part also is GM oi CF; therefore also the remainder MK is the lame part of the remainder FD that the whole GK is of the whole CD. [vii. 7] Again, since, whatever part KH is of CD, the same part also is EL of CF, while CD is greater than CF", therefore HK is also greater than EL. Let KN be made equal to EL . Therefore, whatever part KN is of CD, the same part also is KJV of CF; therefore also the remainder JV/f is the same part of the remainder FD that the whole A'// is of the whole CD, [vn. 7] But the remainder MK was also proved to be the same part of the remainder FD that the whole GK is of the whole CD; therefore also the sum of MK, NH is the same parts of DF that the whole HG is of the whole CD. But the sum of MK, NH is equal to EB, and HG is equal to BA ; therefore the remainder EB is the same parts of the remainder FD that the whole AB is of the whole CD. Q. E. D. It - - «--<( and i==~d. {m <n) then fl-f=^(i-i). Euclid's proof amounts to the following. Take e equal to - b, and /equal to - d. ft H Then since, by hypothesis, b->d. and, by VII. 7, e -f- -{b- d). fi vii. H, 9J PROPOSITIONS 8, 9 309 Repeat this for all the parts equal to t and/that there are in a, b respec- tively, and we have, by addition {a, b containing m of such parts respectively), »'i.:~n = "'-{b-i). But m{i—/) = a-c. Therefore a-e- (d-d). n The propositions v[i, 7, 8 are complementary to v. 5 which gives the corresponding result with multipit in the place of " part " or " parts." Proposition 9. If a number be a pari of a number, and another be the same part of another, alternately also, whatever part or parts ike first is of the third, the same part, or the same parts, will the second also be of the fourth. For let the number A he a part of the number BC, and another, D, the same part of another, £/^, .. , that A is of BC ; I say that, alternately also, whatever part or ^ parts A is of D, the same part or parts is BC , a oi£F3.\so. ^ ^ For since, whatever part A is of BC, the same part also is D of £/^, therefore, as many numbers as there are in BC equal to A, so many also are there in £1^ equal to D. Let BC be divided into the numbers equal to A, namely BG, GC, and ^/^into those equal to D, namely £//, HF\ -w ,„ thus the multitude of BG^ GC will be equal to the multitude of EH, HF. Now, since the numbers BG, GC are equal to one another, and the numbers EH, HF are also equal to one another, while the multitude of BG, GC is equal to the multitude of EH, HF, therefore, whatever part or parts BG is of EH, the same part or the same parts is GC oi HF also ; so that, in addition, whatever part or parts BG is of EH, the same part also, or the same parts, is the sum BC of the sum £F. [vit. s, 6] IflD BOOK VII [vii, 9, to But BG is equal to A, and EH to D\ ' • therefore, whatever part or parts A is of D, the same part or the same parts is BC of EF also. Q. E. D. If a = - b and e = ~ d, then, whatever fraction {" part " or " parts") a is of ft ft c, the same fraction will ihe of ti. Dividing i into each of its parts equal to a, and d into each of its parts equal to i, it is clear that, whatever fraction one of the parts a is of one of the parts c, the same fraction is any other of the parts a of any other of the parts ^. And the number of the parts a is equal to the number of the parts /:, viz. n. Therefore, by vii. 5, 6, na is the same fraction of »c that a is of (, i.e. 6 is the same fraction of d that o is of <:. Proposition 10. // a numier be parts of a number, and another be ike same Paris 0/ another, alternately also, wliatever parts or part the first is of the third, the same parts or the same part will the second also be of t/ie fourth, For let the number AB he parts of the number C, and another, DE, the same parts of another, ^; I say that, alternately also, whatever parts or part AB is of -DE, the same parts or the same part is C of E also. For since, whatever parts AB is o( C, '^' the same parts also is DE of E, therefore, as many parts of C as there are in AB, so many parts also of E are there in DE. Let AB be divided into the parts of C, tiamely AG, GB, and DE into the parts of E, namely D//, HE ; thus the multitude of AG, GB will be equal to the multitude oiDH, HE. Now since, whatever part AG xsoi C, the same part also IsDHo^E, alternately also, whatever part or parts AGx^tS DH, the same part or the same parts is C of E also. [vu. 9] For the same reason also, whatever part or parts GB is of HE, the same part or the same parts is C of J^ also ; v[[. lo, ii] PROPOSITIONS 9—11 JH SO that, in addition, whatever parts or part AB is of /?£', the same parts also, or the same part, is C of F> [vii. 5, 6] Q. E. D. \{ a = — b and e ~ —d, then, whatever fraction a is of c, the same fraction is j of d. To prove this, a is divided into its m parts equal to t>jn, and c into its »J parts equal to djn. Then, by v[i. 9, whatever fraction one of the m parts of a is of one of the m parts of ^, the same fraction is a of d. And, by vii, 5, 6, whatever fraction one of the /// parts of a is of one of the t» parts of c, the same fraction is the sum of the parts of a (that is, o) of the sum of the parts ol c (that is, i). Whence the resliit follows. Fn the Greek text, after the words " so that, in addition " in the last line but one, is an additional explanation making the reference to vii. 5, 6 cl^rer, as follows: "whatever part or parts AG is of DII, the same part or the same parts is GS of HE also ; therefore also, whatever part or parts ^C is of DH, the same part or the same parts is AB of DE also. [vii. 5, 6] But it was proved that, whatever part or parts AG \s of DH, the same part or the same parts is C of Faho ; therefore also " etc, as in the last two lines of the text. Heiberg concludes, on the authority of P, which only has the words in the margin in a later hand, that they may be attributed to Theon. . ■.. Proposition ii. If, as whole is to whole, so is a number subtracted to a number subtracted, the retnainder will also be to the remainder as whole to whole. As the whole ^5 is to the whole C£>, so tet A£ subtracted be to C/^ subtracted ; I say that the remainder £B is also to the remainder FD as the whole AB to the whole CD. Since, as AB is to C£), so is A£ to CF, whatever part or parts AB is of CD, the same part or the same parts is AB of CF a.lso ; [vn. Def. 20] Therefore also the remainder BB is the same part or parts of FD that AB is of CD. [vn, 7, 8] Therefore, as £B is to FD, so is AB to CD. [vn. Def. 30] Q. E. D. It will be observed that, in dealing with the proportions in Props, 11—13, Euclid only contemplates the case where the first number is "a part" or "parts" of the second, vhile in Prop. 13 he assumes the first to be "a part" F jtjl BOOK VII [vii. II, 12 or "parts" of the third also; that is, the first number ts in all three propositions assumed to be less than the second, and in Prop. 13 less than the third also. Yet the figures in Props. 1 1 and 1 3 are inconsistent with these assumptions. If the facts are taken to correspond to the figures in these propositions, it is necessary to take account of the other possibilities involved in the definition of proportion (vii. Def 20), that the first number may also he a multiple, or a multiple //kj "a part" or " parts" (including owe as a multiple in this case), of each number with which it is compared. Thus a number of different cases would have to be considered. The remedy is to make the ratio which is in the lower terms the first ratio, and to invert the ratios, if necessary, in order to make " a part "or " parts " literally apply. If a : i ^ ^ : d, {a > c, b > tf) then {a — c):{b~d) = a;b. This proposition for numbers corresponds to v. 19 for magnitudes. The enunciation is the same except that the masculine (agreeing with apiS/io!) takes the place of the neuter (agreeing with niytdm). The proof is no more than a combination of the arithmetical definition of proportion (vu. Def. 20) with the results of vii, 7, 8. The language of propor- tions is turned into the language of fractions by Def. zo ; the results of vii. 7, 8 are then used and the language retransformed by Def. 20 into the language of proportions. Proposition 12. If there be as Tnany numbers as we please in proportion, then, as one of ike antecedents is to one of the consequents, so are all the antecedents to ail the consequents. Let A, By C, D be as many numbers as we please in proportion, so that, as A is to By so is C to Z? ; I say that, as .^ is to B, so are A, C to B, D. For since, as A is to B, so Is C to D, aI bI c whatever part or parts A is of B, the same part or parts is C q{ D also. [vn. Def. 20] Therefore also the sum of A, C is the same part or the same parts of the sum of B, D that A is of B. [vii. 5, 6] Therefore, as A is to B, so are A, C to B, D. [vn. Def. 20] If a:({ = l,:V = c:i;=..., then each ratio is equal to {a + ^ + ^+ ...) : (0' + *'+/ + ,..). The proposition corresponds to v, 1 2, and the enunciation is word for word the same with that of v. 12 except that apiSfio? takes the place of f^iyt&oi. Again the proof merely connects the arithmetical definition of proportion (vn. Def 20) with the results of vii, 5, 6, which are quotttd as true for any number of numbers, and not merely for two numbers as in the enunciations of VII. s, 6. '3. U] PROPOSITIONS 11—14 313 Proposition 13. If four numbers be proportional, they will also be propor- tional alternately. Let the four numbers A, B, C, D ha proportional, so that, as A is to B, so is C to Z? ; I say that they will also be proportional alternately, so that, as A is to C, so will B be to D. For since, as A is to B, so is C to D, therefore, whatever part or parts A is of B, the same part or the same parts is C of Z? also. [v[i. Def. ao] Therefore, alternately, whatever part or parts A is of C, the same part or the satne parts \5 B oi D also. Therefore, as ^ is to C, so is B to D. If [vii. 10] [y\\. Def. so] Q. E. D. a : b = e : d, then, alternately, a : c = b id. The proposition corresponds to v. 16 for magnitudes, and the proof consists in connecting vii. Def. 20 with the result of vii. 10. Proposition 14. If there be as many numbers as me please, and others equal to them in multitude, which taken two and two are in the same ratio, they will also be in the same ratio ex aequali. Let there be as many numbers as we please A, B, C, and others equal to them in multitude D, E, F, which taken two and two are in the same ratio, so ihat, as .(4 is to B, so is Z? to ^, and, as ^ is to C, so is E to F\ m ■ f . ■ I say that, ex aequali, as A is to C, so also is D to F. "B" D c — f For, since, as A is to B, so is D to E, therefore, alternately, as A is to D, so is B to E. [vn. 13] 314 BOOK Vir [vii. 14, IS Again, since, as B is to C, so is E to F, therefore, alternately, as B is to Ey so is C to F. [vii. 13] But, as J? is to E, so is W to D; therefore also, as y^ is to D, so is C to F. Therefore, alternately, as A is to C, so is D to F. . [iij If a: b = d.e, and d : c = t -.ft tKen, ex aepiali, a ; c = d ./; and the same is true however many successive numbers are so related. The proof is simphcity itself. By VII. 13, alternately, a : il - i : f, and b : e = c '. f. . ■ _ Therefore a\d = c:f, 1 and, again alternately, a : c^d if. Observe that this simple method cannot be used to prove the corresponding proposition for magnitudes, v. 22, although v. 22 has been preceded by the tivo propositions in that Book corresponding to the propositions used here, viz, V. 16 and v, 1 1. The reason of this is that this method would only prove V. 22 for six magnitudes all t>f tht sttmt kind, whereas the magnitudes in v. jj are not subject to this limitation. Heiberg remarks in a note on V[r. 19 that, while Luclid has proved several propositions of Book v. over again, by a separate proof, for numbers, he has neglected to do so in certain cases; e.g., he often uses v. 1 1 in these pro- positions of Book VII., V. 9 in vii. 19, v. 7 in the same proposition, and so on. Thus Heiberg would apparently suppose Euclid "^o use v, 1 1 in the last step of the present proof (Raiies whkk art the same with ike same ratio are also the same with one another). I think it preferable to suppose that Euclid regarded the last step as axiomatic ; since, by the definition of proportion, the first number is the same multiple or the same part or the same parts of the second that the third is of the fourth : the assumption is no more than an assumption that the numbers or proper fractions which arc respectively equal to the same number or proper fraction are equal to one another. Though the proposition is only proved of six numbers, the extension to as many as we please (as expressed in the enunciation) is obvious. Proposition i 5. If an unit measure any number, and another number measure any other number tfie same number of times, alternately also, the unit ivill measure the third number the same number of times that the second measures the fourth. vit. 15] PROPOSITIONS 14. >5 jij For let the unit A measure any number BC, and let another number D measure any other number EF ~^— + !l! — ? the same number of times ; I say that, alternately also, the ^ f L F unit A measures the number D the same number of times that BC measures EF. For, since the unit A measures the number BC the same number of times that D measures /:F, therefore, as many units as there are in BC, so many numbers equal to D are there in ^^also. Let BC be divided into the units in it, BG, GH, HC, and EF into the numbers EK, KL, Z./^ equal to D. Thus the multitude of BG, GH, HC will be equal to the multitude of EK, KL, LF. And, since the units ^C GH, HCzx^ equal to one another, and the numbers EK, KL, LF are also equal to one another, while the multitude of the units BG, GH, HC is equal to the multitude of the numbers EK, KL, LF, therefore, as the unit BG is to the number EK, so will the unit GH be to the number KL, and the unit HC to the number LF, Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents ; [vii. iz] therefore, as the unit BG is to the number EK, so is BC to EF. But the unit BG is equal to the unit A, and the number EK to the number D. Therefore, as the unit A is to the number D, so is BC to EF. Therefore the unit A measures the number D the same number of times that BC measures EF. Q. E. d. If there be four numbers \,m,a, ma (such that i measures m the same number of times that a measures ma), i measures a the same number of times that m measures ma. Except that the first number is unity and the numbers are said to tfieasure instead of being a ^art of others, this proposition and its proof do not differ from VII, 9 J in fact this proposition is a particular case of the other. 31« ■. BOOK VII 1 i" [vii. 1 6 Proposition i6. // two numbers by multiplying one another make certain numbers, the numbers so produced mill be equal to one another. Let A, B he two numbers, and let A by multiplying B make C and ^ by multiplying ,j. A make D ; f^ I say that C is equal to D, b For, since A by multiply- c — ■ ing B has made C, o therefore B measures C ac- — % cording to the units in A. But the unit E also measures the number A according to the units in it ; therefore the unit E measures A the same number of times that B measures C. Therefore, alternately, the imit E measures the number B the same number of times that A measures C. [vn. ij] Again, since B by multiplying A has made D, therefore A measures D according to the units in ^. ' ' But the unit E also measures B according to the units in it ; therefore the unit E measures the number B the same number of times that A measures D. But the unit E measured the number B the same number of times that A measures C ; therefore A measures each of the numbers C, D the same number of times. Therefore C is equal to i?. ' q. e. d. ■1. The numbers >□ produced. The Gieek hu al yoituw i( niriir, " the (numbers) produced /rffjw fJutJt." By *'from them'* Euclid means "from the original numbers," though this is not very clear even in the Greek. I think ambiguity is best avoided by leaving out the words. This proposition proves that, if any nttmbers bt fmtttiplied together, the order of muUiplication is indifferent, ox ab-ba. It '\s important to get a clear understanding of what Euclid means when he speaks of sne number multiplying another, vti. Def, 15 states that the effect of "a multiplying b" is taking a times b. We shall always represent " a times b " by ab and " b times a " by ba. This being premiseidj the proof that ab = ba may be represented as follows in the language of proportions. VII. i6, ij] PROPOSITIONS i6, 17 ^p . ... Ku [vii. 13] By V!i. Def. 10, I : a = i : ai. Therefore, alternately, I : t = a : a&. Again, by vii. Def. 10, X : i = a : ba. Therefore a : ab = a \ ba. or ttb-ba. Euclid does not use the language of proportions but that of fractions or their equivalent measures, quoting vn, 15, a particular case of vii, 13 differently expressed, instead of vii, 13 itself. Proposition 17. If a number by muUiplymg itvo 7iuml>ers make certain numbers, the numbers so produced will have ike same ratio as the numbers multiplied. For let the number A by multiplying the two numbers B, C make D, B\ - ' ' I say that, as .5 is to C, so is D to E. For, since A by multiplying B has made D, therefore B measures D according to the units in A. A B C- F But the unit /^also measures the number A according to the units in it ; therefore the unit F measures the number A the same number of times that B measures D. Therefore, as the unit P is to the number ^, so is ^ to D. [vii. Def. 30] For the same reason, as the unit F is to the number A, so also is C to £■ ; therefore also, as .5 is to /?, so is C to £. Therefore, alternately, as B is to C, so is D to E. [vn. 13] Q. E. D. b K^ab \ ac. In this case Euclid translates the language of measures into that of proportions, and the proof is exactly like that set out in the last note. By VII. Def. so, i : a = b : ai, and X -.a-c: ac. Therefore b : ah^c \ac, and, altematdy, 6:c = ai:at. ' [*"• 'S] $1* BOOK VII ^« [vii. 18,19 ' " ' Proposition 18. ,5.ui-.!jiT // two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers. For let two numbers A, B \yj multiplying any number C make D, E ; I say that, as A is to B, so is D c For, since A by multiplying e C has made Z?, therefore also C by multiplying A has made Z?. [vil 16] For the same reason also ^ ,. ^, C by multiplying B has made E. Therefore the number C by multiplying the two numbers A, B has made D, E. Therefore, as j4 is to B, so is D to E. [vii. 17] It is here proved that a:b=€ie:be. The argument is as follows. ac = ea. [vil, i6] Similarly 6( - cb. And a:i = ca:€bi ["I- 17] therefcne a : b = tu : 6e. • ' '! ■ .l.l-i;' ■.-. I , Proposition 19. If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional. Let A, B, C, Dhe four numbers in proportion, so that, as j4 is to B, so is C to i? ; and let A by multiplying D make £, and let B by multiply- ing C make E; I say that E is equal to E. For let A by multiplying C make G, VII. 19] PROPOSITIONS 18, 19 319 Since, then, A by multiplying C has made G, and by multiplying D has made E, the number A by multiplying the two numbers C, D has made G, E. Therefore, as C is to Z?, so is G to E. [vu. 17] But, as C is to V, sols A 10 B ; therefore also, as ^ is to 5, so is C to E. Again, since A by multiplying C has made G, but, further, B has also by multiplying C made F, the two numbers A, B by multiplying a certain number C have made G, E. Therefore, as A is to B,so\s G to E. ' [vii. 18] But further, as A is to B, ^i is G to E also ; therefore also, as G Is to E, so is G to F. Therefore G has to each of the numbers E, E the same ratio ; therefore E is equal to E. Again, let E be equal to E; I say that, as .^ is to B, so is C to D. For, with the same construction, since E is equal to F, therefore, as (? is to E, so is G to F. But, as £7 is to E, so is C to D, and, as G is to E, so is A to B. Therefore also, as A is to ^, so is C to D. ■ -' ' ' J. y. E. D. ■ . r. I.. ■■ If ■•; I:. a :b = e:d, then ad=kc; and conversely. The proof is equivalent to the following, (i) at : ad—{ : d ■f\ if.xhi •■. I I =a : i, '■•" Bui ^"^ ,W a:»^acii(. Therefore _• ^ i ■ • m : ad = ae : be, or ad=be. [cf. V. 9] ^ ••• a y ■•'l-.v . , [cf. V. 7] [vn. 17] [VII. 18] [vn. 17] '" • (vn. 18] 3M BOOK VII [tu. 19, 10 BOOK VII ) Since ad = b<i. ac \ ad = ae ; be. But ac : ad- e : d. and ac;bc~a : b. Therefore a \b = c \ d. [vii. 17] [VII. iB] As indicated in the note on vii. 14 above, Heiberg regards Euclid t& basing the inferences contained in the last step of part (i) of this proof and in the first step of part (2) on the propositions v, 9 and v. 7 respectivefy, since he has not proved those propositions separately for numbers in this Book. I prefer to suppose that he regarded the inferences as obvious and not needing proof, in view of the definition of numbers which are in pro- portion. E.g., if at is the same fraction (" part " or " parts ") of ad that at is of be, it is obvious that ad must be ei^ual to be. Heiberg omits from his text here, and relegates to an Appendix, a proposition appearing in the manuscripts V, p, ^ to the effect that, if ihret numbers be proportional, the product of the extremes is equal to the square of the mean, and conversely. It does not appear in P in the first hand, B has it in the margin only, and Campanus omits it, remarking that Euclid does not give the proposition about three proportionals as he does in vi. 17, since it is easily proved by the proposition just given. Moreover an-Nairiri quotes the proposition about three proportionals at an obsemaiion on vii. 19 probably due to Heron (who is mentioned by name in the preceding paragraph). Proposition 20. _ The least numbers of those whitk have ike same ratio with them measure those which Jtave the same ratio the same number of times, the greater the greater and t lie less the less. For let CD, EF be the least niimbens of those which have the same ratio with A, B ; I say that CD measures A the same number of times that EF measures B. Now CD is not parts of ^. For, if possible, let it be so ; therefore EF is also the same parts of B that CD is of ^. [vn. 13 and Def. ao] Therefore, as many parts of A as there are in CD, so many parts of ^ are there also in EF, Let CD be divided into the parts of A, namely CG, GD, and £"^into the parts oi B, namely EH, HF\ thus the multitude of CG, GD will be equal to the multitude of EH, HF. o E VII. ao] PROPOSITIONS 19, 20 -3*1 Now, since the numbers CG, GD are equal to one another, and the nunnbers EH, HF are also equal to one another, while the multitude of CG, GD is equal to the multitude of EH, HF, therefore, as CG is to EH, so is GD to HF, ' " " Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents. [vu. u] Therefore, as CG is to EH, so is CD to EF. Therefore CG, EH are in the same ratio with CD, EF, being less than they : which is impossible, for by hypothesis CD, EF are the least numbers of those which have the same ratio with them. Therefore CD is not parts of A ; therefore it is a part of it. [vit, 4] And EF is the same part of B that CD is of ^ ; [vii. 13 and Def. 20] therefore CD measures A the same number of times that EF measures B. Q. E. D. 1( a, b are the least numbers among those which have the same ratio (i.e. if ajb is a fraction in its lowest terms), and t, d are any others in the same ratio,, i.e. if a: b'^t :d, then o = - f and b = - d, where tt is some int^er. ft tt , 1 . _ I The proof is by reductio ad absurdum, thus. [Since a<c,a\% some proper fraction (" part " or "parts ") of e, l^ vii. 4.] Now a cannot be equal to —e, where m is an integer less than n but greater than i. For, if a = -f, b= -rfalso. [vii, 13 and Def. aol Take each of the m parts of a with each of the m parts of b, two and two ; the latio of the members of all pairs is the same ratio ~ a : ■- b. ^ mm Therefore - ix : - 6 = a:b. ' fvit. la] m nt But — a and — b are respectively less than a, b and they are in the same tit tti ratio : which contradicts the hypothesis. 3» BOOK VII [vii. so, ji Hence a can only be " a part " of r, or a is of the form - c, ft and therefore d is of the form - rf. • , .•■ Here also Heibe^ omits a proposition which was no doubt interpolated by Theon (B, V, p, ^ have it as vii. 22, hut P only has it in the margin and in a later hand ; Campanus also omits it) proving for numbers the m aequali proposition when "the proportion is perturbed," i.e. (cf. enunciation of V. jj) if a:b = e\f, (i) and 6:e = d:e, (2) then a •.e = d:/. The proof (see Heiberg's Appendix) depends on vii. 19. From (i) we have of —be, and from (2) bt — cd. [vii. 19] Therefore af= cd, and accordingly a:c-d:f. ,. [vii. 19] Proposition 21. Numbers prime io one another are the least of those which have the same ratio with them, '■ Let A, B be numbers prime to one another; I say that A, B are the least of those which have the same ratio with them. For, if not, there will be some numbers less than A, B which are in the same ratio with A, B. Let them be C, D. Since, then, the least numbers of those which have the same ratio measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent, [vii. 20] therefore C measures A the same number of times that D measures B. Now, as many times as C measures A, so many units let there be in E. Therefore D also measures B according to the units in E. C D I VII. ai, It] PROPOSITIONS 20— »a 313 And, since C measures ^ according to the units in £, therefore £ also measures A according to the units in C. [vn. 16] For the same reason £ also measures B according to the units in D, [vii. 16] Therefore £ measures A, B which are prime to one another : which is impossible. [vn. Def. u] Therefore there will be no numbers less than A, B which are in the same ratio with A, B. Therefore A, B are the least of those which have the same ratio with them. Q. E. D. In other words, \\ a,b are prime to one another, the ratio a : J is " in its lowest terms." The proof is equivalent to the following. If not, suppose that f, </ are the least numbers for which a \b-e \ d. [Euclid only supposes tome numbers f, d in the ratio of o to i such that ir<a, and (consequendy) d-^b. It Js however necessary to suppose that f, d are the least numbers in that ratio in order to enable vn. 3o to be used m the proof.] Then [vn. ao] a = mt, and b = md, where m is some integer. Therefore o = cm, b - dm, [vn. 16] and m is a common measure of a, b, though these ate prime to one another . which is impossible. [vn. Def. la] Thus the least numbers in the ratio of <7 to ^ cannot be less than a, i thetnselves. Where I have quoted vn. i6 Heiberg regards the reference as being to VII. ■ 5. I think the phraseology of the text combined with that of Def. 15 suggests the former rather than the latter. Proposition 22, T/te hast numbers of those which have the same ratio with them are prime to one another. Let .^, ^ be the least numbers of those which have the same ratio with them ; I say that A, B are prime to one g another. c For, if they are not prime to one d another, some number will measure ^ them. Let some number measure them, and let it be C ^ BOOK VII [vii. aa, aj And, as many times as C measures A, so many units let there be in D, „:. -t rw-h and, as many times as C measures B, so many units let there be in E \<s Since C measures A according to the units in D, .<^w'ik therefore C by multiplying D has made A, [vii, Def. 15] For the same reason also C by multiplying E has made B. '• ' ■ Thus the number C by multiplying the two numbers /?, E has made A, B ; therefore, as D is to E, so is ^ x.o B\ [vii. 17] therefore D, E are in the same ratio with A, B, being less than they : which is impossible. Therefore no number will measure the numbers A, B. Therefore A, B are prime to one another. Q. E. U. \i a: b\%" in its lowest terms," a, b are prime to one another. Again the proof is indirect. If a, b are not prime to one another, they have some common measure f. Mid .„ . , a = m^, b = ne. Therefore m : n-a : b. [vii. 17 or iS] But m, n are less than a, b respectively, so that a ; A is not in its lowest terms : which is contrary to the hypothesis. Therefore etc. Proposition 23. .. . ., . > .. , -i-a,-''' If two numbers be prime to one another, (he number which measures the one of them wilt be prime to the remaining number. Let A, B be two numbers prime to one another, and let any number C measure A ; I say that C, B are also prime to one another. For, if C, B are not prime to one another, some number will measure C, B. Let a number measure them, and let it be D. Since D measures C, and C measures A, therefore D also measures A. a a c 6 But it also measures B; vfi. 13, 2i\] PROPOSITIONS 2^—14 3»S therefore Z? measures A, B which are prime to one another : which is impossible, [vii, Def. la) Therefore no number will measure the numbers C B. Therefore C, B are prime to one another. Q. E. D. If a, mil are prime to one another, b is prime to a. For, if not, some number d will measure both a and *, and therefore both a and mb ; which is contrary to the hypothesis. Therefore etc. Proposition 24. If two numbers be prime to any number, their product also will be prime to the same. ' " " For let the two numbers A, B be prime to any number C, and let A by multiplying B make D \ I say that C, D are prime to one another. For, if C, D are not prime to one another, some number will measure C, D. Let a number measure them, and let it be^. Now, since C, A are prime to one another, and a certain number E measures C, therefore A, £ 3.re prime to one another. [v:t. 23] As many times, then, as E measures Z?, so many units let there hem E; therefore E also measures £> according to the units in E. [vii. 16] Therefore E by multiplying E has made 2?, [vit. Def. 15] But, further, A by multiplying B has also made D ; therefore the product of E, E is equal to the product of A, B. But, if the product of the extremes be equal to that of the means, the four numbers are proportional ; [vn. 19] therefore, as £ is to A, so is B to E. '* ^'' But A, £ are prime to one another, numbers which are prime to one another are also the least of those which have the same ratio, [vn. 21J and the least numbers of those which have the same ratio with them measure those which have the same ratio the same jalL BOOK VII [vii. n, a. 5 number of times, the greater the greater iind the less the less, that is, the antecedent the antecedent and the consequent the consequent ; [vn. ao] therefore £ measures B. But it also measures C ; therefore £ measures B, C which are prime to one another : which is impossible. [vn. Def. i*] Therefore no number will measure the numbers C, D. Therefore C D are prime to one another. Q. E, D. I. their product. A ii viitSir ivtbpunt, literal); " the (number) pradoced rrom ihem," will liencefart)) be translated as "tjieir product." If n, b a^ both prime to c, then ab, c are prime to one another. The proof is again by reduciis ad absurdum. If ab, c are not prime to one another, let them be measured by a ^nd be equal to md, mi, say, respectively. Now, since a, c are prime to one another and d measures e, a, d are prime to one anbther. [vn. 33] But, since ab = md, d;a = b:m. [vn. 19] Therefore [vit. *o] d m^sures *, or b =pd, say. . 1 ,. But e = ttd. Therefore d measures both i and c, which are therefore not prime to one another : which is impossible. Therefore etc ■ '■ ' •'■ "" ' ' '" ^;!j Proposition 25. // two numbers bt prime to one another, the product of one of them into itself will be prime to the remaining one. Let A, B be two numbers prime to one another, . ., and let A by multiplying itself make C: „, „ j I say that B, C are prime to one another. For let D be made equal to A. Since A, B are prime to one another, and A is equal to D, therefore D, B are also prime to one another. Therefore each of the two numbers D, A is prime to B ; therefore the product of D, A will also be prime to B. [vn. 34] vii, IS, 16] PROPOSITIONS 24—26 ' J»J But the number which is the product of D, A is C Therefore C, B are prime to one another. q. e. d. I. the product of one of them into Itself. The Greeks h in raO hfht a6r«iw ytr&furott literatlj ^'the number produced from the one of them/' leaves '* multiplied into it$«lf '^ to be understood. If a, i are prime to one another, - v ■ ■ .' >' ■ ' • a' is prime to i. v 1 - 1 Euclid takes d equal to a, so that d, a are both prime to ^. Hence, by vii. 24, da, i.e. a', is prime to i. The proposition is a particular case of the preceding proposition ; and the method of proof is by substitution of different numbers in the result of that proposition. ' Proposition 26. If two numbers ie prime to two numbers, both to each, their- products also will be prime to one another. For let the two numbers A, B he prime to the two numbers C, D; both to each, and let A by multiplying B ^^ q make E, and let C by multi- g t, plying Z* make ./^; I say that E, F are prime to p one another. For, since each of the numbers A, B is prime to C, therefore the product oi A, B will also be prime to C. [vn. *4] But the product of A, B is E ; > therefore B, C are prime to one another. , , For the same reason ^, - m..-; /. Et D are also prime to one another. •., . .-n- u Therefore each of the numbers C, D is prime to E. Therefore the product of C, D will also be prime to E. [vii. *4] But the product of C, D is F. Therefore E, Fa.Tc prime to one another. q. e. d. If both a and i are prime to each of two numbers e, d, then ai, cd will be prime to one another. Since a, b are both prime to c, ab, t are prime to one another. [vn. 34] Similarly ab, d are prime to one another. Therefore c. d are both prime to ab, and so theiefoK is ed- [vn. 34] j(*8 BOOK VII [vii. 27 Proposition 27. If two numbers be prime to one another, and each by multiplying itself make a certain number, the products V}iU be prime to one another; and, if the original numbers by multi- plying the products make certain numbers, the latter will also be prime to one another [and this is always the case with the extremes\ Let A, B be two numbers prime to one another, let A by multiplying itself make C, and by multiplying C make D, and let B by multiplying itself make E, and by multiplying E make F; I say that both C, E and D, F are prime to one another. For, since A, B are prime to one another, and v4 by multiplying itself has made C, therefore C, B are prime to one another. [vn. 25] Since then C, B are prime to one another, and B by multiplying itself has made B, therefore C, E are prime to one another. [«£} Again, since A, B are prime to one another, and B by multiplying itself has made E, therefore A, E are prime to one another. \id:\ Since then the two numbers A, C are prime to the two numbers S, E, both to each, therefore also the product of A, C is prime to the product of B, E. [vn. 26] And the product of A, C is D, and the product of B, E is F. Therefore D, F are prime to one another. Q. E. D. If a, b are prime to one another, so are a', ^ and so are a*, ^; and, generally, a", i" are prime to one another. The words in the enunciation which assert the truth of the proposition for any powers are suspected and bracketed by Heiberg because (i) in jripl rots Bitpout the use of ojtpoi is peculiar, for it can only mean " the last products," and _(*) the words have nothing corresponding to them in the proof, much less ts the generalisation proved. Campanus omits the words tn the enuncia'^ vii. *7, a8] PROPOSITIONS 37, 18 3*9 tion, though he adds to tht proof a remark that the proposition is true of any, the same or different, powers of «, i. Heiberg concludes that the words are an interpolation of date earlier than Theon. Euclid's proof amounts to this. Since a, i are prime to one another, so are a*, * [vii. a si. and therefore also a', ff. [vii, as] Similarly [vii. as] «, ^ are prime to one another. Therefore a, a* and b, *■ satisfy the description in the enunciation of VII. a 6. Hence a', ^ are prime to one another. .■ . • r > ^ > ...,..,,, Proposition 28. If two numbers be prime to one another, the sum will also be prime to each of tkem ; and, if the sum of two numbers be prime to any one of them, ike original numbers will also be prime to one another. For let two numbers AB, BC prime to one another be added ; I say that the sum AC \s also prime a" ■ ~S 6 to each of the numbers AB, BC. D For, if CA, AB are not prime to .^ , one another, some number will measure CA, AB. Let a number measure them, and let it be Z). Since then D measures CA, AB, therefore it will also measure the remainder BC. ,^ ^^ | But it also measures BA ; therefore D measures AB, BC which are prime to one another : which is impossible. [vii, Def. u] Therefore no number will measure the numbers CA, AB; therefore CA, AB are prime to one another. For the same reason AC, CB are also prime to one another. Therefore CA is prime to each of the numbers AB, BC. Again, let CA, AB he prime to one another ; I say that AB, BC are also prime to one another. For, if AB, BC are not prime to one another, some number will measure AB, BC. 330 BOOK VII [vn. zS, 19 Let a number measure them, and let it be />. Now, since Z? measures each of the numbers AB, BC, it will also measure the whole CA. But it also measures AB ; therefore D measures CA, AB which are prime to one another: which is impossible. [vii. Def. la] Therefore no number will measure the numbers AB, BC. Therefore AB, BC are prime to one another. Q. E. D. If a, b are prime to one another, a-¥6 will be prinm to both a and b ; and conversely. For supjwse (a + *), a are not prime to one another. They must then have some common measure d. Therefore d also divides the difference {a + li) - a, or b, as well as a ; and therefore a, h are not prime to one another : which is contrary to the hypothesis. Therefore a + i is prime to a. Similarly b + * is prime to #. , , , 1, The converse is proved in the same way. Heibei^ remarks on Euclid's assumption that, if c measures both a and b, it also measures a±b. But it has already (vji. i, j) been assumed, more generally, as an axiom that, tn the case supposed, c measures a±J>b. Proposition 29. : ..j - . 1 > Any prime number is prime to any number which it does not measure. Let ^ be a prime number, and let it not measure B ; I say that B, A are prime to one another. For, if B, A are not prime to one ^a another, — ^b some number will measure them. c Let C measure them. Since C measures jff, and A does not measure B, therefore C is not the same with A. Now, since C measures B, A, therefore it also measures A which is prime, though it is not the same with it : which is impossible. vn. S9, 30) PROPOSITIONS iS— .10 $$i Therefore no number will measure B, A. Therefore A, B are prime to one another. Q. E. D. If a is prime and does not mtsasure b, tht:n a, b axa priiiiu lo onu anuthur. The proof is self-evident. Proposition 30. If two numbers by muliiplying one another utake some number, and any prime number measure the product, it will also measure one of the original numbers. For let the two numbers A, B by multlplvirig one another make C, and let any prime number D measure C ; ^ , I say that D measures one of the b_ numbers A, B. c ■ For let it not measure A. x) Now D is prime ; e therefore A, D are prime to one another. [vu. 29J And, as many times as D measures C, so many units let there be in .£". Since then D measures C according to the units in E, therefore D by multiplying E has made C. [vk. IJef, 15] Further, A by multiplying B has also made C\ therefore the product of D, E is equal to the product of A, B. Therefore, as /? is to ^, so is .5 to E. [vu, 19] But D, A are prime to one another, primes are also least, [vu. zi] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent ; , -, [vu. 20] therefore D measures B. Similarly we can also show that, if D do not measure B, it will measure A. Therefore D measures one of the numbers A, B. Q. E. u. 332 '.'ir BOOK VII [vii. 30, 31 If IT, a primt; number, measure ai, e will measure either a ur i. Suppose c does not measure a. Therefore ^, a are prime to one another. [vii, 19] Suppcrae ab-me. Therefore e:a-b:m. .- 1; ^ ■' . 'i.i. >'< [vu. 19] Hence [vii. JO, 2i] fmeasuresi, ' •■-'• " Similarly, if c does not measure l>, it measures c. Therefore it measures one or other of the two numbers a, k '■.■• •t'' 'i S PkOVOHITION 31. • • ^•''• Any compost ie number is measured by some prime number. Let ^ be a composite number ; . . ' I say that A is measured by some prime number. For, since A is composite, ' ' S some number will measure it A Let a number measure it, and let it b- — be B. c— Now, if B is prime, what was en- joined will have been done. 10 But if it is composite, some number will measure it Let a number measure it, and let it be C Then, since C measures B, and .5 measures .^, ■•:-,••• .■•''■ • • ■ ' therefore C also measures >(4. ■■ IS And, if C is prime, what was enjoined will have been done. But if it is composite, some number will measure it Thus, if the investigation be continued in this way, some prime number will be found which will measure the number a) before it, which will also measure A. For, if it is not found, an infinite series of number will measure the number A, each of which is less than the other; which is impossible in numbers. Therefore some prime number will be found which will as measure the one before it, which will also measure A. Therefore any composite number is im^asured by some prime number. V". 3>— 33] PROPOSITIONS 30—33 333 S. if B Is prime, what was enjoined vrlll have been done, i.e. the implied prsiiim of finding n prime number which measures A* iS. some prime number will be found which will measure. In the Creek the sentence stops here, but it U necessary to add the words " the number before it, which will also measure j4," which are found a few lines further down. It is possible that the words ma^ have fidlen out of P here by a simple mistake due to biitnari\ivtae (Heiberg). Heiberg relegates to the Appendix an alternative proof of this proposition, to the following effect. Since A is composite, some number will measure it. Let B be the least such number. I say that 3 is prime. For, if not, B is composite, and some number will measure it, say C; so that C is less than B. But, since C measures S, and B tneasures A, C must measure A. And C is less than B -. which is contrary to the hypothesis, • '• " Proposition 32. '' "' ' ■ Any number either is prime or is measured by some prime number, • ■. .-. t-j.i .V 1-. . • • Let .^4 be a number; I say that A either is prime or is measured by some prime number. , , If now A IS prime, that which was a '■ enjoined will have been done. But if it is composite, some prime number will measure it [vii. 31] Therefore any number either is prime or is measured by some prime number. Q, E. D, Proposition 33, Gwen as many numbers as we p/ease, to find the least of those which have the same ratio with them. Let A, B, C be the given numbers, as many as we please ; thus it is required to find the least of s those which have the same ratio with A, B, C. * A, B, C are either prime to one another or not. Now, \i A, B, C are prime to one I I 10 another, they are the least of those y^ I which have the same ratio with them, [vii. 3i] But, if not, let D the g^-eatest common measure of .^, B, C be taken, [vii. 3] B C E 1- t 1 1 Q 334 BOOK VII IDJtn , [VI,. 33 and, as many times as D measures the numbers A, B, C IS respectively, so many units let there be in the numbers £, F, G respectively. Therefore the numbers E, F, G measure the numbers A, B, C respectively according to the units in D. \yu. 16] Therefore E, E, G measure A, B, C the same number of 30 times ; therefore E, E, G are in the same ratio with A, B, C. [vii. Drf. 30] I say next that they are the least that are in that ratio. For, if E, E, G are not the least of those which have the same ratio with A, B, C, 'S there will be numbers less than E, F, G which are in the same ratio with A, B, C. Let them )x. H, K, L ; .-..-. therefore H measures A the same number of times that the numbers K, L measure the numbers B, C respectively. 30 Now, as many times as H measures A, so many units let there be in M; therefore the numbers K, L also measure the numbers B, C r^pectively according to the units in M. And, since H measures A according to the units in M, 35 therefore M also measures A according to the units in H. [vn. 16] For the same reason M also measures the numbers B, C according to the units in the numbers K, L respectively ; ; - Therefore M measures A, B, C. 40 Now, since H measures A according to the units in M, therefore H by multiplying M has made A. [vn. Def. 15] For the same reason also ,i •. E by multiplying D has made A. Therefore the product oi E, D\& equal to the product of 4S H,M. Therefore, as ^ is to H, so is M to D. [vn. 19] But E is greater than H ; therefore M is also greater than D, It And it measures A, B, C\ VII. 33j PROPOSITION 33 38S JO which is impossible, for by hypothesis D is the greatest common measure of A, B, C. Therefore there cannot be any numbers less than E, F, G which are in the same ratio with A, B, C. Therefore F, F, G are the least of those which have the js same ratio with A, B, C. •rviT i.iii'w •i^,\tTr.m nefH ".n* ! r-;( uJ ^^jwt.,- Q- E. D. 17. the numbeis E, F, C measure the numbers A, B, C respectively, literally (as usual) "eacli of the numl>er» £, F, G measures each of the numbers A, — rr Given any numbets a, h, c, .,,, to find the least numbers that are in the same ratio. Euclid's method is the obvious one, and the result is verified by reductio ad absurdum. We wi!t, like Euclid, take three numbers only, a, h, c. Letf, their greatest common measure, be found [vn. 3], and suppose that a = mg, i.e. gm, , v> . . [vn. 16] 6 = ug, U.g«. ,j^^ ,. ': = />i. I.e. ^. X , , •.•.„■-; It follows, by vn. Def. 20, that ' • v 1 . 4 1 ^ t .-* n :[w H m : H ■.p = a : i :e, j . . i «, », / shall be the numbers required. For, if not, let x, y, e be the least numbers in the same ratio as a, b, r, being less than /«, n, p. Therefore a = kx (or xk, vn. 16), i> = ky {at yk), \ i,l -^S c = kz (or zk), • where k is some integer. [vii. ao] Thus «lj{*= a = xk. Therefore "'^■'i ' '' "• m:x = k:g. .fj rj'.! [vii. 19] And m^ x; therefore k-> g. Since then k measures a, A, c, it follows that g is not the greatest comaon measure ; which contradicts the hypothesis. Therefore etc. "^ ^■'- ^-ifr-'I'T It is to be observed thai Euclid merely supposes that x, y, g are smaller numbers than m, n, p in the ratio of a^b, c\ but, in order to justify the next inference, which apparently can only depencl on vn. ao, x, y, t must also be assumed to be the hait numbers in the ratio of «, b, c. The inference from the last proportion that, since m> x, i >^is supposed by Hetberg to depend upon vit. 13 and v. 14 together. I prefer to regard Euclid as making the inference quite independently of Book v. E.g., the proportion could just as well be written X : m=g : k, ''^^ when the definition of proportion in Book vn. (Def. 20) gives all that we want, since, whatever proper fraction x is of m, the same proper fraction is g of k. ^ BOOK VII [vii. 34 Proposition 34. Given two numbers , to find the least number which th^ measure. Let A, B be the two given numbers ; thus it is required to find the least number which they measure, Now^,^ are either prime to one ^ B another or not. ^ First, let A, B be prime to one ^ another, and let A by multiplying B make C; ^ ^ therefore also B by multiplying A has . made C, : . • [vil 16] Therefore A, B measure C I say next that it is also the least number they measure. For, if not, A, B will measure some number which is less than C Let them measure D. , „ . Then, as many times as ^ measures D, so many units let there be in £, and, as many times as B measures D, so many units let there be in f; therefore A by multiplying £ has made D, and B by multiplying /^ has made Z? ; [vii. Def, 15] therefore the product of ^, j6" is equal to the product of ,5, F. Therefore, as A is to B, so is /^ to £. [vn, 19] But A, B are prime, primes are also least, [vn, ti] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less ; [vn. 10] therefore B measures B, as consequent consequent. And, since A by multiplying B, E has made C. D, therefore, as B is to E, so is C to Z>. [vn. 17] But B measures E \ therefore C also measures D, the greater the less : which is impossible. Til. 34] PROPOSITION 34 337 Therefore A, B do not measure any number less than C ; therefore C is the least that is measured by A, B. Next, let ^, B no*" be prime to one another, and let F, E, the least numbers of those which have the same ratio with A, B, be taken ; [vil 33] therefore the product of -^, ^ is equal to the product of ^, F. [vii. 19] And let A by multiplying E "" m,i«7>- , make C ; * b therefore also B by multiplying F p e has made C ; therefore A, B measure C. d •' ,, I say next that it is also the least h number that they measure. For, if not. A, B will measure some number which is less than C. Let them measure D. And, as many times as A measures D, so many units let there be in G, and, as many times as B measures D, so many units let there be in H. Therefore A by multiplying G has made D, and B by multiplying H has made D. Therefore the product of A, G is equal to the product of B,H; therefore, as A is to B, so is H to G. [vil 19] I, But, as y4 is to B, so is F to E. ,^^i, ..■ Therefore also, as /^ is to £, so is li to G. ^,^^ But F, E are least, ,j_^, , _ and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less ; ^ ^ [vn. to] therefore E measures G. And, since A by multiplying E, G has made C, /?, therefore, as £ Is to G, so is C to D. [vn. 17] But £ measures G ; itj v .•»». therefore C also measures D, the greater the less : which is impossible. ^ BOOK VII [vii. 34 Therefore A, B will not measure any number which is less than C. Therefore C is the least that is measured by A, B. Q, E. D. This is the problem of finding the itait common multipk of two numbers, as a, b. .. , . I, If «, ^ be prime to one another, the l.cm. is ab, , For, if not, let it be rf, some number less than aA Then d-ma-nb, where wi, n are int^ers. Therefore a.b-n:m, [vil. rg] and hence, a, b being prime to one another, b measures m. [vii. ao, »i} But b\M = iA;aM [vti. 17] = ab:d. Therefore ah measures d: which is impossible. ' TT. If a, i be not prime to one another, find the numbers which are the least of those having the ratio of a to b, say «, « ; [vit. 33] then a: b^m -.n, and an-bm (=f, say); [vir, 19] e is then the i.c.m. For, if not, let it be i/ (< c), so that ap-bq = d, where/, q are integers. Then a:6 = q:p, [vii. 19] whence m : n = q -.p, so that n measures/. [vii. 20, ai] And n : p = an • ap = c : d, 90 that e measures d: which is impossible. ' " Therefore etc. By VII 33> m n 4 ~~S _b ^ s Hence the I.C.M . is ab S , where £ is the g.c.m. of a, b. vii. 35> 36] propositions 34—36 339 Proposition 35. // two numbers measure any number, ike least number measured by tkem will also measure the same. For let the two numbers A, B measure any number CD, and let E be the least that they measure; * p I say that E also measures CD. For, if E does not measure CD, let E, measuring DF, leave C/^less than itself. Now, since A, B measure E, •'• and E measures DF, therefore A, B will also measure DF. But they also measure the whole CD ; therefore they will also measure the remainder CF which is less than E: ^ which is impossible. - '. Therefore E cannot fail to measure CD ; ,-i- ,.., -.j therefore it measures it. Q. E. D. The koii common multiple of any two numbers must measure any other common multiple. The proof is obvious, depending on the fact that, if any number divides » and b, it also divides a -ph. _i n. ; Proposition 36, Given three numbers, to find the least number which they measure. Let A, B, C be the three given numbers ; thus it is required to find the least . , number which they measure, a '"' .' Let D, the least number mea- b sured by the two numbers A, B, c — be taken. [vn. 34] d- Then C either measures, or e I'l does not measure, D. , . •_, First, let it measure it. .$4a • BOOK VII [VII. 36 But A, B also measure D\ ■■■ . .-- - . - — t therefore A, B, C measure D. I say next that it is also the least that they measure. For, if not, A, B, C will measure some number which is less than D. Let them measure E. Since A, B, C measure E, ' therefore also A, B measure E. Therefore the lease number measured by A, B will also measure E. [vn. 35] But D is the least number measured by ^, Z? ; therefore D will measure E, the greater the less : which is impossible. Therefore A, B, C will not measure any number which is less than D ; therefore D is the least that A, B, C measure. Again, let C not measure D, ' and let E, the least number measured by . C, D, be taken. [vn. 34] Since A, B measure D, q and D measures E, ' therefore also A, B m^sure E, 1 But C also measures E ; ' - ' ^ therefore also A, B, C measure E, 1 say next that it is also the least that they measure. For, if not. A, B, C will measure some number which is less than E. Let them measure E. _;.rs Since A, B, C measure E, therefore also ^, ^ measure /^; therefore the least number measured by A, B will also measure E. [vn. 35] But D is the least number measured by A,*B ; therefore D measures F. , .,v •-; But C also measures /^; ejiii-^i.v ..'•'. therefore A C measure ./% so that the least number measured by D, C will also measure E. VII- 3<S, 37] PROPOSITIONS 36. 37 3*^ But £ is the least number measured by C, D; therefore £ measures J^, the greater the less : which is impossible. Therefore A, B, C will not measure any number which is less than £. Therefore £ is the least that is measured by A, B, C Q. K. D. Euclid's rule for finding the ucm. of ihrte numbers a, j, ^ is the rule with which we are familiar. The L.CI1. of a, b is first found, say d, and then the L.C.H. of d and c is found. Euclid distinguishes the cases (i) in which c measures d, (i) in which c does not measure d. We need only reproduce the proof of the general case (3). The method is that of rtducfio ad a&surdum. Let e be the L.C.M, of d, c. Since a, i both measure d, And d measures «, '' ' "^ ' a, b both measure e. So does c. Therefore e is iome common multiple of a, b, c. If it is not the ioj^, let/be the L.CM. Now a, b both measure/; therefore d, their L.aM., also measures/ [vu, 35] Thus d, ( both measure/ therefore e, their l.c.m,, measures/: : ,, . •• [vii. 35] which is impossible, since /< e. Therefore etc. The process can be continued ad libitum, so that we can find the L.C.M., liot only of threes but of as many numbers as we please. Proposition 37. ' '■ If a number be measured by any number, the number which is measured will have a part called by the same name as the measuring number. For let the number A be measured by any number B \ I say that A has a part called by the same name as B, a For, as many times as B measures A, b so many units let there be in C. ^ Since B measures A according to the p units in C, and the unit D also measures the number C according to the units in it, ■^ XI / BOOK VII m [VII. 37, gS therefore the unit D measures the number Cthe same number of times as B measures A. Therefore, alternately, the unit D measures the number B the same number of times as C measures A ; [vii. 15] therefore, whatever part the unit D is of the number B, the same part is C of -^ also. But the unit D is a. part of the number B called by the same name as it ; therefore C is also a part of A called by the same name as B, so that A has a part C which is called by the same name as B. Q. E. D. If 6 measures a, then 7 th of ii is a whole number. Let a-m.b. Now m = m.x. Thus It m, b, a satisfy the enunciation of vti. 15 ; therefore m measures a the same number of times that i measures b. But I is T th part of b ; '' theiefore w is r th part of a. 1 .- 'i^, Proposition 38. ;r. // a number have any part whatever, it will be measured by a number called by the same name as the part. For let the number A have any part whatever, B, and let C be a number called by the same name as the part B ; I say that C measures A. * ' For, since ^ is a part of A called by ^ the same name as C, ^ and the unit D is also a part of C called by the same name as it, therefore, whatever part the unit D is of the number C, the same part \s B o( A also ; therefore the unit D measures the number C the same number of times that B measures A. viL 38, 39j PROPOSITIONS 37—39 343 • Therefore, alternately, the unit D measures the number B the same number of times that C measures A. [vit. 15] Therefore C measures A. n I . J >i/i 3d' Q. E. D. This proposition is practically a. restatement of the preceding proposition. It asserts that, if * is - th part of «, ,._,., i.e.,if <5 = -a, . J -^ ^ '-'i ! m . . , then m measures a, .131-11, J i? « '' i I' .ri A We have . .; . b=-~ a, . •, ■1- m and !=—«(. Therefore i, m, i, a, satisfy the enunciation of vit. 15, and thet«foTe m measures n the same number of times as i measures i, or I ' , n( = 1 a. . ._. . . 1- r. i ■ - '" ' '•• " Proposition 39. ill To find the number which is the least that will have given parts. Let A, B, C be the given parts ; thus it is required to find the number which is the least thai will have the parts A, B, C. A B c, D Let D, £, F be numbers called by the same name as the parts A, B, C, and let G, the least number measured by D, E, 7^ be taken. [vii. 36] Therefore G has parts called by the same name as D, E, F. [vn. 37] But A, B, C are parts called by the same name as Z>, E, F\ therefore G has the parts A, 3, C. I say next that it is also the least number that has. 344 BOOK VII [viL 39 For, if not, there will be some number less than G which will have the parts A, B, C. Let it be H. m - :n' * Since H has the parts A, By C, therefore H will be measured by numbers called by the same name as the parts A, B, C. fvu. 38] But D, E, F are numbers called by the same name as the parts A, B, C\ therefore H is measured by D, E, F. And it is less than G : which is impossible. Therefore there will be no number less than G that will have the parts A, B, C. Q. E. D, This again is practically a restatement in another form of the problem of finding the L.C.M. To find a number which has - th, t th and - th parts. Let d be the l.c.m. of a, d, c. Thus d has -eh, rth and -th parts. [vti. 37] If it is not the least number which has, let the least such number be <■. Then, since e has those parts, e is measured by a, 6,e; and e<d: which is impossible. BOOK VIII. Proposition i. ■r»f // there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the numbers are the least of those which have the same ratio with them. Let there be as many numbers as wk please, A^ B, C, D, in continued proportion, and let the extremes of them a- ~ g — A, D\x. prime to one another; b f I say that A, B,C, D are the Q least of those which have the " h same ratio with them. For, if not, let E, F, G, H h^ less than A, B, C, D, and in the same ratio with them. Now, since A, B, C, D are in the same ratio with E, F. G,H, and the multitude of the numbers A, B, C, D is equal to the multitude of the numbers E, F, G, H, therefore, ex aequali, as A is to D, so is E to H. [vii. 14] But A, D are prime, primes are also least, [vii. ai] and the least numbers measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent. [vn. ao] 346 BOOK Vin [vni. i, i Therefore W measures £, the greater the less : which is impossible. Therefore E, F, G, H which are Jess than A^ B, C, D are not in the same ratio with them. Therefore A, B, C, D are the least of those which have the same ratio with them. Q. E. D. What we call a geometrical progression is with Euclid a series of terms "in continued proportion " (ifijt o'lutAo^}. This proposition proves that, if a, f, <:,... ji are a series of numbers in geometrical progression, and if o, k are prime to one another, the series is in the lowest terms jxj^ible with the same common ratio. The proof is in femi by redudio ad absurdum. We should no doubt desert \.\\\&form while retaining the substance. If «', iJ', c', . . . A' be any other series of numbers in c.p. with the same common ratio as before, we have, tx atquali, a : k = a' : k', [vii. 14] whence, since a, k are prime to one another, a, k measure a', k' respectively, so that a', k' are greater than a, k respectively. Proposition 2. To find numbers in continit^d proportion, as many as may be prescribed, and Ike least that are in a given ratio. Let the ratio of ^4 to ^ be the given ratio in least numbers ; thus it is required to find numbers in continued proportion, as many as may be prescribed, and the least that are in the ratio of A to j9. -^0 Let four be prescribed ; let A by multiplying itself make C, and by multiplying B let it make D \ let B by multiplying itself make E ; further, let A by multiplying C, D, E make F, G, H, and let B by multiplying E make K. VIII. 2] PROPOSITIONS 1,1 ^j Now, since A by multiplying itself has made C, " *' and by multiplying B has made /?, therefore, as A is to B, so is C to /?. [vii. 17] Again, since A by multiplying B has made D, and ^ by multiplying itself has made £, therefore the numbers A, B hy multiplying £ have made the numbers D, E respectively. • 'r'l Therefore, as ^ is to B, so is D to E. [vii. 18] But, as y4 is to .5, so is C to Z? ; therefore also, as C is to D, so is D to E. And, since A by multiplying C, D has made F, G, therefore, as C is to D, so is F to C [vn. 17] But, as C is to D, so was A to B\ therefore also, as ^ is to B, so is F to G, Again, since A by multiplying /?, ^ has made G, //, therefore, as ZJ is to E, so is G to //'. [vii, 17] But, as i? is to £", so is /4 to B. Therefore also, as A is to B, so is G to H. And, since .^4, ^ by multiplying E have made H, K, therefore, as A is to B, so is H to ^. [vii, 18] But, as A is to B, so is /^ to G, and (7 to jfiT. Therefore also, as F is to G, so is G to H, and H io K; therefore C, Z?, £", and Z^ G, H, K are proportional in the ratio of A to B. I say next that they are the least numbers that are so, For, since A, B are the least of those which have the same ratio with them, and the least of those which have the same ratio are prime to one another, , ,,1 [vn. aa] therefore A, B are prime to one another. And the numbers A, B hy multiplying themselves re- spectively have made the numbers C, E, and by multiplying the numbers C, E respectively have made the numbers F, K\ therefore C, E and F, A'are prime to one another respectively, [vn. 27] But, if there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another. 348 BOOK VIII [vin. a, 3 they are the least of those which have the same ratio with them. [viii. i] Therefore C, D, E and F, G, H, K are the least of those which have the same ratio with A, B. q. e. d. PokisM. From this it is manifest that, if three numbers in continued proportion be the least of those which have the same ratio with them, the extremes of them are squares, and, if four numbers, cubes. To find a series of numbers in geometrical progression and In the least terms which have a given common ratio (understanding by that term ilu ratio of one term to the next). Reduce the given i^tio to its lowest terms, say, a : i. (This can be done by VII. 33.) Then a", a*-'i, a'-^lr', ... a'*"-', u^-', i- is the required series of numbers if {« + i ) terms are required. That this is a series of terms with the given common ratio is clear from vit. 17, i8. That the G.P. is in the smallest terms possible is proved thus. - a, 6 are prime to one another, since the ratio a : i is in its lowest terms. [vii. 2»] Therefore o", #" are prime to one another ; so are «", ^ and, generally, *", ^. _ [vil. tj] Whence the g.p. is in the smallest possible terms, by viu. i. The Porism observes that, if there are h terms in the series, the extremes are («- i)th powers. ',• .■• . 'iV) Proposition 3. ,^ ;\ 1^ '• , .v .'t // as many numbers as we please in continued proportion be the least of those which have the same ratio mith them, the extreines of them are prims to one anot/ter. Let as many numbers as we please, A, B, C, D, m con- tinued proportion be the least of those which have the same ratio with them ; ■i-i < — E — F :> . • 1 — O H K -L M N VIII. 3] PROPOSITIONS i, 3 349 I say that the extremes of them A, D are prime to one aaother. For let two numbers E, F, the least that are in the ratio o^ A, B, C, D, be taken, [vii. 33] then three others G, H, K with the same property ; and others, more by one continually, [vm. 2] until the multitude taken becomes equal to the multitude of the numbers A, B, C, D. Let them be taken, and let them be L, M, N, 0. Now, since E, F are the least of those which have the same ratio with them, they are prime to one another, [vti. 22] And, since the numbers E, F by multiplying themselves respectively have made the numbers G, K, and by multiplying the numbers G, K respectively have made the numbers L, O, [vin. z, For.] therefore both G, /f and L, O are prime to one another, [vii. 27] And, since A, B, C, D are the least of those which have the same ratio with them, while Z,, Mi N, O are the least that are in the same ratio with A, B, C, D, and the multitude of the numbers A, B, C, D is equal to the multitude of the numbers L, M, N, O, therefore the numbers A, B, C D are equal to the numbers Li M, N, O respectively ; therefore A is equal to L, and D to O. And Z., O are prime to one another, • Therefore A, D are also prime to one another. Q, E, D, The proof consists in merely equating the given numbers to the terms of a series found in the manner of viii. 2. \i a, b,c, ... k {n terms) be a geometrical progression in the lowest terms having a given common ratio, the terms must respectively be of t>e form found by viii. 2, where a : j9 is the ratio a • 6 expressed in its lowest terms, so that a, J8 are prime to one another [vn. 21], and hence «""', fi*~^ are prime to one another [vii. 27], ., . But the two series must be the same, so that « = «■-', * = ;8"-' 350 BOOK VIII [viit. 4 Proposition 4. Given as many ratios as we please in leasi numbers, to find numbers in continued proportion which are ike least in the given ratios. Let the given ratios in least numbers be that of A to B, s that of C to D, and that q{ E to F\ thus it is required to find numbers in continued proportion which are the least that are in the ratio of A to B, in the ratio of C to D, and in the ratio of E to F, A— B ■•' D E F «— 5 H M !i P L Let G, the least number measured by B, C, be taken. in And, as many times as B measures G, so many times also let A measure ^, and, as many times as C measures G, so many times also let D measure /C, Now E either measures or does not measure K. 15 First, let it measure it. And, as many times as E measures A*, so many times let E measure L also. Now, since A measures // the same number of times that jff measures G, *> therefore, as y^ is to B, so is // to G. [vn. Def. io, vil 13] For the same reason also, as C is to D, so is G to K, and further, as ^ is to /^, so is A' to Z. ; therefore If, G, K, L are continuously proportional in the «s ratio of A to B, in the ratio of C to D, and in the ratio of E XoF. I say next that they are also the least that have this property. viii. 4] PROPOSITION 4 3S> For, if H, G, K, L are not the least numbers continuously 30 proportional in the ratios of A to B^ of C to D, and of B to F, let them be N, 0, M, P. :- • j »., : Then since, as A is to B, so is N to O, while ^, ^ are least, and the least numbers measure those which have the same 35 ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent ; therefore B measures 0. [vn. ao] ' , .1 For the same reason 40 C also measures O; ;• therefore B, C measure O ; therefore the least number measured by B, C will also measure O. [vii. 35] But G is the least number measured by J9, C ; 45 therefore G measures O, the greater the less : which is impossible. Therefore there will be no numbers less than H, G, K, L which are continuously in the ratio of A to B, of C to D, and oiBxaF. Sf> Next, let £ not measure J^. A B Q — E F .,, , 1 , 0- H t 1 < K — M N — P — Let M, the least number measured by B, K, be taken. And, as many times as K measures M, so many times let H, G measure A', respectively, and, as many times as B measures M, so many times let F ss measure P also. Since H measures A'' the same number of times that G measures O, therefore, as /T is to 6^, so is A^ to O, [vn. 13 and Def, so] 3S2 BOOK VIII [VIII. 4 But, as ^ is to 6^, SO is /4 to ^ ; 60 therefore also, as A is to B, so is A'^ to O. 1 io 1 . 1 ■• For the same reason also, as C is to D, so is O to M. Again, since ^ measures ^ the same number of times that F measures P, 65 therefore, as ^ is to /^ so is ^ to /* ; [vu. 13 and Def, zo} therefore A'^, O, M, P are continuously proportional in the ratios of A to B, of C to /?, and of E to F, I say next that they are also the least that are in the ratios A:B, C:D. E:F. 70 For, if not, there will be some numbers less than A^, O, M, P continuously proportional in the ratios A\B, C.D, E:F , , Let them be Q, R, S, T. Now since, as Q is to R, so is A to B, 75 while A, B arc least, and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent, [vii. jo] therefore B measures R. 80 For the same reason C also measures R ; therefore B, C measure R. Therefore the least number measured by B, C will also measure R. [vii. 35] But G is the least number measured by ^, C\ is therefore G measures R. And, as G is to R, so is A" to ,$" : [vii. 13J therefore K also measures S. But E also measures S; therefore E, K measure S, 90 Therefore the least number measured by E, K will also measure S. [vn. 35] But M is the least number measured by E] K \ therefore M measures 5, the greater the less : which is impossible. 9S Therefore there will not be any numbers less than A', O, Mt P continuously proportional in the ratios of A to B, of C to D, and of E to F; VIII. 4] PROPOSITION 4 353 therefore N, O, M, P are the least numbers continuously proportional in the ratios A:B,C:D,E:F. q. e. d. So, 71, 09< the ratios A : B, C : D, B : F. TKu mbbreruted expression is in the Greek dI AB, TA, EZ \V- The terid " in continued proportion " is here not used in its proper sense, since a geometrical progression is not meant, but & series of terms each of which b^rs to the succeeding term a given, but not the same, ratio. The proposition furnishes a good example of the cumbrousness of the Greek method of dealing with non-determinate numbers. The proof in fact is not easy to follow without the help of modem symbotical notation. If this be used, the reasoning can be made clear enough. Euclid takes three given ratios and therefore requires to ^tsAfour numbers. We will leave out the simpler particular case which he puts first, that nameljr in which B accidentally measures K, the multiple of D found in the first few lines ; and we will reproduce the general case with Mr« ratios. Let the ratios in their lowest terms be ; ^ a:i,e:d,e:/ .Tjiir Take li, the l.c.u. of i, e, and suppose that , ; \ ' , .' /, = mi — ne. Form the numbers ma, mli \, rut, 1 . . ,•.■ = nc\ ■ I ■ These are in the ratios of ji to d and of <r to rf respectively. ■.)•'• 'I Next, let /, be the l.c,m. of nd, e, and let ; ,'. J i 4 =/«rf = qe. Now form the numbers pma, pmb \ , ptid \ , qf^ =pfie f =ge I and these are the four numbers required. If they are not the least in the given ratios, let f' y> ^' « , >\ ... ^^. V. .l.o'->i1| be less numbers in the given ratios. _. , Since ii : i is in its lowest terms, and '"''"''■ '-""' •' '"■ a \ b~x : y^ i measures V. ,h , . Similarly, since e : a=^ : z, t measures >■, Therefore /j, the L.C.M. of l>, f, measures >. But /, xnd\~e:d\^y: Therefore nd measures z. And, since e :f~z ; u, e measures z. Therefore 4i the L.C.M. of nd, e, measures z : which is impossible, since i</j or pnd. The step (line 86) inferring that G : H = K ; S 'n of course alternando from G:K[=C: D\ = Ji . S. It will be observed that viii. 4 corresponds to the portion of vi, 33 which shows how to compound two ratios between straight lines. >i. i-{'A -:> ti.H X. ■ .-•ll.ii [ r^'i ■ 'h 354 BOOK VIII [viii. 5 Proposition 5, Plane numbers have to one another the ratio compounded of the ratios of their sides. Let -4, ^ be plane numbers, and let the numbers C, D be the sides of A, and E, F oi B\ s I say that A has to B the ratio com- g pounded of the ratios of the sides. -. ^ For, the ratios being given which C _e — p has to E and D to F, let the least q numbers G, HyKih^t are continuously ^ 10 in the ratios C\E,D:Fh^ taken, so ^ that, ^ as C is to E, so is G to Hy and, as D is to F, so is H to K. [viii. 4] And let D by multiplying E make L. IS Now, since D by multiplying C has made A, and by multiplying £" has made L, therefore, as C is to -£", so is ^ to Z,. ,. [vii. ij] But, as C is to ^, so is (7 to //^ ; therefore also, as G is to H, so is A to L. 20 Again, since E by multiplying Z? has made L, and further by multiplying F has made ^, therefore, as /? is to /s so is Lkq B. - [vii. 17] But, as Z? is to F^ so\% H to K\ therefore also, as H is to K, so is L to ^. *s But it was also proved that, as fz is to H, so is ^ to Z. ; therefore, ex aeguali, as G^ is to K, so is A to B. [vn. 14] But G has to K the ratio compounded of the ratios of the 30 sides ; therefore A also has to B the ratio compounded of the ratios of the sides. Q. e. d. I, 5, 19, 31. compounded of the ratios of Ibeir sides. As in v[. 13, the Greek has the less exact phitue, " cornpounded of their sides." If a = ed, b = tf, then a has to b the ratio compounded oi c le and d :/ Take three numbers the least which are continuously in the given ratios. viii. 5, 6] PROPOSITIONS 5, 6 JSS If / is the L.CII. of e, d and l=mt = nd, the three numbers are m/:, me \, nf. [vill. 4] ~nd) Now dc:dt=c\c [vii. 17] = mc : me - me \ nd. Also td:tf=d\f [vii. 17] = ttd\nf. Therefore, ft* atguali, cd •.ef=mc : nf = (ratio compounded of 1: . e and d ;/). It will be seen that this proof follows exactly the method of vi. 23 for parallelograms. Proposition 6. // there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will any other measure any other. Let there be as many numbers as we please. A, B, C, D, E, in continued proportion, and let A not measure B ; I say that neither will any other measure any other. -F — a H Now it is manifest that A, B, C, D, E do not measure one another in order ; for A does not even measure B. I say, then, that neither will any other measure any other. For, if possible, let A measure C. And, however many A, B, C are, let as many numbers F, G, H, the least of those which have the same ratio with A, B, C, be taken. [vn. 33] Now, since F, G, H are in the same ratio \-ith A, S, C, and the multitude of the numbers A, B, C is equal to the multitude of the numbers F, G, H, therefore, ex aequali, as A is to C, so is F to H. [vil 14] 3jfir BOOK VIII . , [viri. 6, 7 And since, as A is to ^, so is 7^ to G, a - ■ • ' ■ ■»> while A does not measure B, therefore neither does F measure G ; [vu. Def. ao] therefore F is not an unit, for the unit measures any number. "^om F, H are prime to one another. [vm. 3] And, as F is to H, so is ^ to C ; therefore neither does A measure C Similarly we can prove that neither will any other measure any other. Q. E. D. Let a^b,c...k\x,^ geometrical progression in which a does not measure h. Suppose, if possible, that a measures some term of the series, as / Take x,y, x, u, v, w the itcat numbers in the ratio a, b, c, d, e,f. Since x ■.y = a:b, and a does not measure b, ' X does not measure ^; therefore x cannot be unity. And, ex aequali, x : w = a :/. Now X, w are prime to one another. [viii. 3] Therefore a does not measure/ We can of course prove that an intermediate term, as b, does not measure a later term / by using the series b, e, d, e, f and remembering that, since b -.c-a; b, b does not measure c. Proposition 7. If there be as many numbers as we please in continued proportion, and the first measure the last, it tvUl measure the second also. Let there be as many numbers as we please, A, B, C, D, in continued proportion ; and let A measure D ; * I say that A also measures B, ^ For, if A does not measure ^' B, neither will any other of the ° — —" numbers measure any other. [vm. 6] But A measures D. Therefore A also measures B. ■," Q. £. D. An obvious proof by redudia ad absurdum from vm. 6, VHI. 8] PROPOSITIONS 6—8 ^^. I. Proposition 8. • ^' If between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between them in continued proportion, so many will also fall in con- tinued proportion between the numbers which have the same ratio with the original numbers. Let the numbers C, D fall between the two numbers A, B in continued proportion with them, and let E be made in the same ratio to ^ as ^ is to ^ ; I say that, as many numbers as have fallen between A., B in continued proportion, so many will also fall between E, F in continued proportion. A e c M O N B F- ~- G ""■ K L ' ■ - - " For, as many as A, B, C, D are in multitude, let so many numbers G, H, K, L, the least of those which have the same ratio with A, C, D, B, be taken ; [vn, 33] therefore the extremes of them (9j L are prime to one another. [vin. 3] Now, since A, C, D, B are in the same ratio with G, H, K,L, and the multitude of the numbers A, C, D, B is equal to the multitude of the numbers G, H, K, L, therefore, ex aequali, as A is to B, so is G to L. [vii. 14] But, as .(4 is to B, so is E to E; therefore also, as G is to Z, so is ^ to A ' But G, L are prime, "'~ primes are also least, [vn. si] and the least numbers measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent. [vn. ^o] 3Sg BOOK VIII [vui. 8, 9 Therefore G measures S the same number of times as L measures F. Next, as many times as G measures E, so many times let /f, K also measure M, N respectively ; therefore <7, H, K, L measure E, M, N, F the same number of times. Therefore G, H, K, L are in the same ratio with E, M, N, F, [vn, Def. 20] But G, H, K, L are in the same ratio with A, C, D, B ; therefore A, C, D, B are also in the same ratio with E, M, N, F , But A, Ct D, B are in Continued proportion ; therefore E, M, N, FAre also in continued proportion. Therefore, as many numbers as have fallen between A, B in continued proportion with them, so many numbers have also fallen between E, F in continued proportion, Q, E. D. t. blL The Gi«k word is iiarhmir, " Ml in " = "can be interpolited." If a:6 = e:/, and between a, b there are any number of geometric means t, d, there will be as 'many such means between f, /. Let 0, fi, y, ■•■, S be the least possible terms in the same ratio as a, c, d,...b. Then o, 8 are prime to one another, [vtu. 3] and, ex atquali, a:% = a\b = €:f. Therefore t = ima,ys: m%, where m is some int^er. [vii. 10] t Take the numbers ma, mjS, my, ... mS. This is a series in the given ratio, and we have the same number of geometric means between ma, mS, or e,/, that there are between a, b. Proposition 9. If two numbers be prime to one another, and numbers fall between them in continued proportion, then, however many numbers fall between them in continued proportion, so many will also fall between each of them and an unit in continued proportion. Let A, B be two numbers prime to one another, and let C, D fall between them in continued proportion, and let the unit E be set out ; I say that, as many numbers as fall between A, B in con- VIII. 9] PROPOSITIONS 8, 9 399 tinued proportion, so many wi]l also fall between either of the numbers A, B and the unit in continued proportion. For let two numbers F, Gy the least that are in the ratio of A, C, D, B, be taken, three numbers H, K, L with the same property, and others more by one continually, until their multitude is equal to the multitude of ^, C, D, B. [vni. a] A H ic • ■ •' D L • B E- F- Q— n -1 . P Let them be taken, and let them be M, N, O, P. It is now manifest that F by multiplying itself has made H and by multiplying // has made M, while G by multiplying itself has made L and by multiplying L has made P. [viii. 3, Por.] And, since M, N, O, P are the least of those which have the same ratio with F^ G, and A, C, D, B are also the least of those which have the same ratio with F, G, [viii. i] while the multitude of the numbers M, N, C, P is equal to the multitude of the numbers A, C, D, B, therefore M, N, O, P are equal to A, C, £>, B respectively ; therefore Af is equal to A, and P to B. ,,. Now, since Fhy multiplying itself has made //, -■■'-■^ therefore F measures // according to the units in F. But the unit F also measures F according to the units in it; therefore the unit F measures the number F the same number of times as F measures If, Therefore, as the unit F is to the number F, so is F to ff. [vii. Def. »o] Again, since F by multiplying // has made M, therefore // measures M according to the units in F, v«.> 36o BOOK VIII [viii. 9, lo But the unit £ also measures the number F according to the units in it ; therefore the unit B measures the number F the same number of times as // measures M. Therefore, as the unit £ is to the number F,so\s// to M. But it was also proved that, as the unit F is to the number F, so is /^ to /^ ; therefore also, as the unit F is to the number F, so is F to If, and // to M. But Jtf is equal \ja A \ therefore, as the unit E is to the number /", so is F to H, and H to A. For the same reason also, as the unit E is to the number G, so is C to Z and L to B. Therefore, as many numbers as have fallen between A, B in continued proportion, so many numbers also have fallen between each of the numbers A, B and the unit E in continued proportion, Q. E. D. Suppose there are n geometric means between a, b, Iwo numbers prime to one another ; there are the same number (n) of geometric means between i and a and between i and b. If c, d... are the n means between a, b, a, i, d ... b are the least numbers in that ratio, since a, b are prime to one another, [viii. ij The terms are therefore respectively identical with a"*', a-^, d'-'jS' ... Q^, iS"*', where o, ^ is the common ratio in its lowest terms. [vni. a. For.] Thus tf = <i"+', * = ^«+>. Now I Lffl = a I a' = a' : q'... =a» :a"*', and I:j8 = j3:^ = ^:j9'...=j8-;j8-*-; whence there are « geometric means between i, a, and between i, b. Proposition to. If numbers fall between each of two numbers and an unit in continued proportion, however many numbers fall between each of tkem and an unit in continued proportion, so many also will fall between the numbers themselves in continued proportion. VIII. lo] PROPOSITIONS 9, lo 361 For let the numbers D, E and F, G respectively fall between the two numbers A, B and the unit C in continued proportion ; I say that, as many numbers as have fallen between each of the numbers A, B and the unit C in continued proportion, so many numbers will also fall between A, B in continued pro- portion. For let D by multiplying F make H, and let the numbers D, F by multiplying Ji make K, L respectively. .■^ i.^' c — D— E — F Q A- H K L I ' t Now, since, as the unit C is to the number D.sois D to £, therefore the unit C measures the number D the same number of times as D measures E. [vn. Def. 10] But the unit C measures the number £f according to the units in D ; therefore the number D also measures E according to the units inZ?; therefore I? by multiplying itself has made E. Again, since, as C is to the number D, so is E to A, therefore the unit C measures the number £> the same number of times as E measures A, But the unit C measures the number D according to the units in D ; therefore E also measures A according to the units in D ; therefore D by multiplying E has made A. - ' ' For the same reason also F by multiplying itself has made G, and by multiplying G has made B. And, since D by multiplying itself has made E and by multiplying F has made //, therefore, as Z> is to F, so is E to If. [vil 17] 3«a BOOK Vm [vin. lo For the same reason also, -: .. ' as D is to /% so is // to C • ' " [vii. i8] Therefore also, as £ is to N, so h N to G. Again, since D by multiplying the numbers £, H has made A, K respectively, therefore, as E is to H, so is ^ to A'. ' [vii. 17] But, as E is to H, so\s D 10 E; therefore also, as D is to E, so is A to K. Again, since the numbers D, E by multiplying H have made K, L respectively, therefore, as D is to F, so is K to L, [vn. 18} But, as D is to F, ^q 'v& A to K \ therefore also, as ^4 is to K, so is K to L. Further, since F by multiplying the numbers //, G has made L, B respectively, therefore, as H is to G, so is L to B. [vil 17] But, as // is to G, so is D to A; therefore also, as /? is to 7% so is Z, to A • ' ■• ' • ' But it was also proved that, as D is to E, so is ^ to ^ and K to L\ therefore also, as A is to K, so is A" to Z and L to B. Therefore A, K, L, B are in continued proportion. Therefore, as many numbers as fall between each of the numbers A, B and the unit C In continued proportion, so many also will fall between A, B\u. continued proportion. Q. E, D, If there be n geometric nutans between i and a, and also between i and i, there will be « geometric means between a and b. The proposition is the converse of the preceding. The n means with the extremes form two geometric series of the fonn I, (I, a' ... a", 0"+', I, j8, ;8'...^, /8-+'. where a"^ ' = n, j3"+' = b. By multiplying the last term in the first line by the first in the second, the last but one in the first line by the second in the second, and so on, we get the series and we have the n means between a arid b. It will be observed that, when EucUd says " Fsr tke sami naion also, as i? is to /J so is .ff to G,'' the reference is really to vn. iS inst^id of vii. 17. viii. lo, ii] PROPOSITIONS lo, II 363 He infers narady that Dx F: Fx F=D : F. But since, by vn, 16, the order of multiplication is indifferent, he is practically justified in saying " for the same reason." The same thing occurs in later propositions. Proposition ii. Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the sieU. , , Let A, B\y& square numbers, " ' and let C be the side of W, and D o( B; I say that between A, B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to D. a For let C by multiplying D make E. b ~ Now, since .<^ is a square and C is o □ its side, £ therefore C by multiplying itself has ,^ made A. . . . ' -i For the same reason also • • t' D by multiplying itself has made B. Since then C by multiplying the numbers C, D has made A, E respectively, therefore, as C is to /?, so is ^ to E, , [vii. 1 7] For the same reason also, as C is to /?, so is ^ to ^. • [vn, iS] Therefore also, as A is to E^ so is E to B. Therefore between A, B there is one mean proportional number. I say next that A also has to B the ratio duplicate of that which C has to D, For, since A, E, B are three numbers in proportion, therefore A has to B the ratio duplicate of that which A has to E. [v. Def. 9] But, as A is to E, so is C to D, Therefore A has to B the ratio duplicate of that which the side C has to D. q. e. d. According to Nicomachus the theorems in this proposition and the next, that two squares have one geometric mean, and two cubes two geometric means, betweerf them are Platonic. Cf. TimamSt 32 a sqq. and the note thereon, p. 294 above. '$64 BOOK VIII [viii. II, 13 a*, ^ being two squares, it is only necessary to form the product ai and to prove that a", ai, ^ are in geometrica) progression. Euclid proves that ^ '. ai = ai : ff' by means of vii. 17, 18, as usuaL In assuming that, since a* is to ^ in the duplicate ratio of a* to at, a* is to P in the duplicate ratio of a to 6, Euclid assumes that ratios which are the duplicates of equal ratios are equal. This, an obvious inference from V. 33, can be inferred just as easily for numbers from vii. 14. '.1 ;i>lr *n» "'J J Jirf iWli. Proposition 12. Between two cube numbers there are two mean proportional numbers, and tlie cube has to the cube the ratio triplicate of that which the side has to the side. Let A, B\>^ cube numbers, and let C be the side of A, and D oi B; I say that between A, B there are two mean proportional numbers, and A has to B the ratio triplicate of that which C has to /?. A- ttt K ' '.''' C — ■A D K .» For let C by multiplying itself make £", and by multiplying D let it make F; let D by multiplying itself make G, and let the numbers C, D by multiplying F make H, K respectively. Now, since .^ is a cube, and C its side, - ... « •« li and C by multiplying itself has made E, therefore C by multiplying itself has made E and by multiply- ing E has made A, For the same reason also D by multiplying itself has made G and by multiplying G has made B. And, since C by multiplying the numbers C, D has made E, /^respectively, therefore, as C is to D, so is E to F, .... ■ [vu. ij] VIII. li, 13] PROPOSITIONS 11—13 3*5 For the same reason also, as C is to /?, so is .^ to G. [vii. iSJ Again, since C by multiplying the numbers E, F has made A, H respectively, therefore, as E is to F, so is A to H, ' [vii. 17] But, as >£' is to F, so is C to /?. Therefore also, as C is to D, so is A to If. Again, since the numbere C, D by multiplying F have made H, K respectively, therefore, as C is to D, so is H to K, [vii. 18] Again, since D by multiplying each of the numbers F, G has made K, B respectively, therefore, as F is to G, so is K to B. [vii, 17] But, as /^ is to G, so is C to Z? ; therefore also, as C is to /?, so is Ava H,H to K, and K to B. Therefore H, K are two mean proportionals between A, B. 1 say next that A also has to B the ratio triplicate of that which C has to D, For, since A, H, K, B are four numbers in proportion, therefore A has to B the ratio triplicate of that which A has to H, [v. Def. 10] But, as y^ is to Ht so is ^ to Z* ; therefore A also has to B the ratio triplicate of that which C has to D. Q. E. D, The cube numbere if, fi being given, Euclid forms the products n^b, a^ and then proves, as usual, by means of vii, 17, 18 that are in continued proportion. He assumes that, since a* has to ^ the ratio triplicate of o* : <^b, the ratio a' : ^ is triplicate of the ratio a : b which is equal to a* : (?b. lliis is again an obvious inference from vii. 14. Proposition 13. If there be as many numbers as we please in continued proportion, and each by multiplying itself make some number, the products ■mill be proportional ; and, if the original numbers by multiplying the products make certain numbers, the latter will also be proportional. 0St Tt- BOOK VIII [viii. 13 Let there be as many numbers as we please, A, B, C, in continued proportion, so that, as -(4 is to Ji, so is /? to C; let A, B, Chy multiplying themselves make D, E, F, and by multiplying D, E, /^let them make G, H, K\ I say that D, E, F and G, H, K are in continued proportion. A a B H K M _ N p Q >• For let A by multiplying B make L, • • .'i ''•• and let the numbers A, B by multiplying L make M. N respectively. And again let B by multiplying C make O, ' ''■ and let the numbers B, C by multiplying O make P, Q respectively. Then, in manner similar to the foregoing, we can prove that D, L, E and G, M, N, H are continuously proportional in the ratio of A to B, and further £, O, F and H, P, Q, K are continuously propor- tional in the ratio of B to C. Now, as A is to B^ so is ^ to C; therefore D, L, E are also in the same ratio with E, O, F, and further G, M, N, H in the same ratio with H, P, Q, K. And the multitude of D, L, E is equal to the multitude of E, O. F, and that of G, M, N, H to that of H, P,Q,K; therefore, ex acgualt, as D is to -£", so is £■ to F, '''' ' and, as tr is to H, so is H to K. [vn. 14] VIII. 13, 14] PROPOSITIONS 13, 14 3«J U a,i, c ...he a. series in geometrical progression, then J J 13 J f ^''^ *l^ '" geometncal progression. Heiberg brackets the words added to the enunciation which extend the theorem to any powers. The words are "and this always occurs with the extremes " (jtai a<i -rtpi tov( ixpov; toGto tnififlairu). They seem to be rightly suspected on the samt: grounds as the same words added to the enunciation of viL 27. There is no allusion to them iii the proof, much less any proof of the extension. Euclid forms, besides the squares and cul^es of the given numbers, the products ad, t^fi, atf, be, t^c, ic\ When he says that " we prove in manner similar to the foregoing," he indicates successive uses of vii. 17, iS as in VIII. 13. With our notation the prcx)f is as easy to sec for any powers as for squares and cubes. To prove that n", ^, <*,.. are in geometrical progression, " "'"' ' Form all the means between a', *", and set out the series a*, a'-'/', a'-'lr ... air-', ff. The common ratio of one term to the next is a ; /S. Next take the geometrical progression *". tr-^ i'-V ... //c^-\ i", v the common ratio of which i& i : c. Proceed thus for all pairs of consecutive terms, ' ' Now a : 6 = fi : c= ... Therefore any pair of succeeding terms in one scries are in the same ratio as any pair of succeeding terms in any other of the series. ,., And the number of terms in each is the same, namely (h + i). Therefore, m atgiialiy a-:*- = i» :(■ = <:" :rf" = ... Proposition 14. 1/ a square measure a square, the side will also measure the side ; and, if the side measure the side, the square will also measure the square. Let A, Bh& square numbers, let C, D be their sides, and let A measure B; I say that C also measures D. A For let C by multiplying D make E ; fl therefore A, E, B are continuously pro- — ^ " portional in the ratio of C to D. [vm. nj e And, since A, E, B are continuously proportional, and A measures B, ' "■ therefore A also measures E. '• • '"- [vm, 7] 3«8 BOOK VIII [viiL 14, IS And, as A is to E, so is C-io D\ ' "" ' therefore also C measures Z?, [vii. Def, ao] Again, let C measure D\ • " I say that A also measures B. For, with the same construction, we can in a similar manner prove that A^ E, B are continuously proportional in the ratio of C to Z?. And since, as C is to D, so is A to E^ and C measures D, therefore A also measures E. [vii. Def. ao] And A, E, B are continuously proportional ; therefore A also measures B. Therefore etc. Q. E. D. If «* measures ^, a measure l> ; and, if « measures b, a* measures P. (1) *i', ab, i° are in continued proportion in the ratio of a to b. (viii. 7] Therefore, since 0' measures ^, ..,-_. a' measures ab. But d^ : ab = a \ b. Therefore a measures b. (2) since a measures b, a* measures ab. And a', ab, P are continuously proportional. Thus ab measures ^, And a' measures ab. Therefore «' measures i". It will be seen that Euclid puts the last step shortly, saying that, since (j^ measures ab, and a*, ab, ^ are in continued proportion, a* measures #*. The same thing happens in viii. 15, where the series of terms is one more than here. Proposition 15. If a cube number measure a cube number, the side will also measure the side ; and, if the side measure the side, the cube will also measure the cube. For let the cube number A measure the cube B, and let C be the side of A and D oi B \ I say that C measures D. VIII. is] propositions 14, IS 3«^ For let C by multiplying itself make E, and let D by multiplying itself make G ; further, let Cby multiplying D make f, '"*» --tt .-•;, . >'> and let C, D by multiplying F make H, K respectively, A o— ' ■'■; K ■' E — ' °ir~ "---^ ■' Now it is manifest that E, F, G and A, H, K, B are continuously proportional in the ratio of Cto D. [viii. 11, u] And, since A, H, K, B are continuously proportional, and A measures B, therefore it also measures H. c- > , ■ [""■ '] And, as y4 is to /f, so is C to Z? ; 1 ,. .. therefore C also measures D. [vii, Dct ao] Next, let C measure D ; '" "" " •^''' "^ '' _ I say that A will also measure B. For, with the same construction, we can prove in a similar manner that A, //, K, B are continuously proportional in the ratio of C to D. And, since C measures D, and, as C is to D, so is A to H, therefore A also measures H^ [vii, Def. ao] so that A measures B also, Q. E. D. If 0* measures V, a measures b \ and via versa. The proor is, mutatii mutandis, the same as for squares. (i) (f,a'6,ai^,i^tK continuously proportional in the ratio of a to * ; and a* measures ^. Therefore «• measures a*i ; [viii. 7] and hence a measures i, "'■ -fi. ••>(_■ 1 (i) Since a measures i, a* measures ci*j. And, 0*, a*3, aJ*, ^ being continuously proportional, each term measures the succeeding term ; therefore a* measures ^. .Ml 1« VI. Sf9 BOOK VIII [viii. i6, 17 -TV ^ Proposition 16. i^^^ If a square numder do not measure a square number, neither will the side measure the side ; and, if the side do not measure the side, neither will the square measure the square. Let A, B be square numbers, and let C, Z? be their sides ; and let A not measure B ; I say that neither does C measure D. * For, if C measures D, A will also ^ measure B. [vni. 14] c But A does not measure B ; d therefore neither will C measure D. Again, let C not measure Z? ; . . ' I say that neither will A measure B. For, if A measures B, C will also measure D. [vin. 14} But C does not measure D ; therefore neither will ^ measure B. Q. E. D. If a' does not measure ^, a mil not measure b; and, if a does not measure h t? will not measure ^. The proof is a mere rtduciio ad absurdum using vin. 14. Proposition 17. " If a cube number do not measure a cube numher, neither will the side measure the side ; and, if the side do not measure the side, neither will the cube measure the cube. For let the cube number A not measure the cube number B, and let C be the side of A, and D a^ of^; 1 say that C will not measure D. For if C measures Z), A will also measure B. [viii. 15] But A does not measure B ; therefore neither does C measure D, Again, let C not measure D ; I say that neither will A measure B. ^ — — D viiL 17, 18] PROPOSITIONS 16—18 371 For, \{ A measures B, C will also measure D. [vm. 1 5] But C does not measure D ; therefore neither will A measure B. Q. E, D. If a* does not measure ^, a will not measure i ; and viee versa. Proved by reducHo ad absurdum employing viii. 15, Proposition 18. Between two similar plane numbers there is one mean proportional number ; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side. Let A, B be two similar plane numbei^, and let the numbers C, D be the sides of A, and E, F of B. K ' O B D E , . . F Now, since similar plane numbers are those which have their sides proportional, [vii. Def. «i] therefore, as C is to D, so \% E x.o F. ' ■ '' ' I say then that between A, B there is one mean propor- tional number, and A has to B the ratio duplicate of that which C has to E^ or D to F, that is, of that which the corre- sponding side has to the corresponding side. Now since, as C is to D, so is E to F, therefore, alternately, as C is to E, so is D to F. [vn, 13} And, since A is plane, and C, D are its sides, therefore D by multiplying C has made A. For the same reason also E by multiplying F has made B. Now let D by multiplying E make G. Then, since D by multiplying C has made A, and by multiplying E has made G, therefore, as C is to E, so is A to G. [vn. 17] 37a " BOOK VIII [viii. i8 But, as C is to £", SO is Z? to /^; • " ' therefore also, as /? is to J^, so is A to G. ''*' Again, since £ by multiplying D has made G, and by multiplying F has made B, therefore, as Z> is to /% so is C to B. '" \ " [vii. 17] But it was also proved that, as Z' is to F, so\% A to G; therefore also, as A is to (7, so is G to B. Therefore A, G, B are in continued proportion. '■•'• Therefore between A, B there is one mean proportional number. •^ I say next that A also has to B the ratio duplicate of that which the corresponding side has to the corresponding side, that is, of that which C has to £" or Z> to F. For, since A, G, B are in continued proportion, A has to B the ratio duplicate of that which it has to G. [v. Def. 9] And, as A is to G, so is C to E, and so is D to F. Therefore A also has to B the ratio duplicate of that which C has to £■ or Z? to F. Q. E. D, \i ab, <rf be " similar plane numbers," i.e. products of factors such that a • b = c ; d, thciie is one mean proportional between ai and at; and ab 'n to ed m the duplicate ratio of a to 1: or of ^ to ^. Fonn the product be (or ad, which is equal to it, by vii. 19). Then ' ab, be] , cd ab, be\, = ad)' is a series of terms in geometrical progression. For a : b=c •■ d. Therefore a:c = b:d. [v". 13] Therefore ab : be = be : cd. ' [vil. 17 and 16] Thus be (or ad) is a geometric mean between ah, cd. ' , And ab is to<rfin the duplicate ratio of ab to be or of be to <rf, that is, of d to f or of ^ to d. vm. 19] PROPOSITIONS 18, 19 jyj Proposition 10, Between two similar solid numbers there fall two mean proportional numbers; and the solid number has to the similar solid number the ratio triplicate of that which the corresponding side has to the corresponding side. Let j4, B he two similar solid numbers, and let C, D, E be the sides of A, and F, G, H of B. Now, since similar solid numbers are those which have their sides proportional, [vu. Def. si] therefore, as C is to /?, so is Fxa G, ••: I'l : '; and, as D is to ^, so Is 6^ to ff. I say that between A, B there fall two mean proportional numbers, and A has to B the ratio triplicate of that which C has to F, D to G, and also E to H. ■ '■' \ . ■■•■ A- B C- F- N- 0- o- E— H K— I ■ M For let C by multiplying D make K, and let F by multiplying G make X. ■> Now, since C, D are in the same ratio with F, G, and /C is the product of C, V, and L the product of F, O, K, L are similar plane numbers ; [vn, Def. ai] therefore between K, L there is one mean proportional number. . , . [vin. 18] • Let it be ^ • ' '" - " ' ' Therefore M is the product of D, F, as was proved in the theorem preceding this. [vm. 18] Now, since D by multiplying C has made K, and by multiplying ^ has made M, therefore, as C is to /^ so \^ K Xa M. [vn. 17] But, as K is to M, so is M to L. Therefore K, M, L are continuously proportional in the ratio of C to F, f^4 . ot .t BOOK VIII [viii. 19 And since, as C is to D, so is F to G, alternately therefore, as C is to /% so is Z? to G. [vii. 13] For the same reason also, as D is to G, so is E to H. Therefore K, M, L are continuously proportional in the ratio of C to F^ in the ratio of D to G, and also in the ratio of E to H. Next, let E, H by multiplying M make A^, O respectively. Now, since ^ is a solid number, and C, D, E are its sides, therefore E by multiplying the product of C, D has made A. But the product of C, D is K; • . ■ - lis.- ; therefore E by multiplying K has made A. • ' • m= For the same reason also ' ' H by multiplying L has made B. Now, since E by multiplying A' has made A, and further also by multiplying M has made N^ therefore, as K is to M, so is j4 to N. [vii. 17] But, as K is to ^, so is C to F, D to £7, and also E to //■; therefore also, as C is to Z', Z? to G, and EtoH,ia\%A to A'^, Again, since £", ^ by multiplying M have made A^, O respectively, therefore, as ^ is to //, so is N to O. [vii. 18] But, as ^ is to H, so is C to /^ and D Xa G\ therefore also, as C is to /% Z? to G, and E to H,^a\% A to iV and A^ to a Again, since //^ by multiplying ^has made f?, and further also by multiplying L has made B, therefore, as ^ is to Z,, so is f? to B, [vil 17] But, as ^ is to Z, so is C to F, D to G, and ^ to H. Therefore also, as C is to F, D to G, and E io H, so not only is t? to ^, but also A to N and A'' to O. Therefore A, N,0, B are continuously proportional in the aforesaid ratios of the sides. I say that A also has to B the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of the ratio which the number C has to F, or D to G, and also E to H. VIII. 19, ao] PROPOSITIONS 19, 20 375 For, since A, N, O, B are four numbers in continued proportion, therefore A has to B the ratio triplicate of that which A has to N, [v. Def. ro] But, as y^ is to A'', so it was proved that C is to F, D to (?, and also E to H. Therefore ^ also has to ^ the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of the ratio which the number C has to F, D to G^ and also E to H, Q. E. D. In other words, M a:b : c=d : e :/, then there are two geometric means between abc, def; and abc is to def in the triphcate ratio of a to d, or b to e, or c \,of Euclid first takes the plane numbers ab, dt (leaving out e, f) and foroos the product bd. Thus, as in viu. 18, ab, bd\ , de -eaj are three terms in geometrical progression in the ratio of a to d, or of i to e. He next forms the products of f,/ respectively into the mean bd. Then afic, ebd, fid, def are in geometrical progression in the ratio of a to ^ etc. Fm abc bd fbd : cbd^ab ; bd=a : d' •.fbd = c:f -. [vii. 17] •.def=bd:de=b:e ] And a : d-b : t = c -.f. The ratio of abc to def is the ratio triplicate of tliat of abc to ebd, ie. of that of a ta d etc. Proposition 20. If <me mean proportional number fall between two numben, the numbers will be similar plane numbers. For let one mean proportional number C fall between the two numbers A, B\ 5 1 say that A, B are similar plane numbers. Let D, E, the least numbers of those which have the same ratio with A, C, be taken ; [vii. 33] therefore D measures A the same number of times that B measures C. [vu. no] 10 Now, as many times as D measures A, so many units let there be in ^; r therefore F by multiplying Z? has made A, so that A is plane, and D, F are its sides. 3.f6 BOOK VIII 'Vi [viii. ao Again, since />, £ are the least of the numbers which have IS the same ratio with C, B, therefore D measures C the same number of times that E measures B. [vn. 20] A' O B- F- Q- As many times, then, as E measures B^ so many units let there be in G\ ao therefore E measures B according to the units in G^; therefore G by multiplying E has made B. ' " Therefore B is plane, and E, G are its sides, Therefore A, B are plane numbers. I say next that they are also similar. - » «S For, t since F by multiplying D has made At and by multiplying E has made C, therefore, as D is to E, so is A to C, that is, C to B. [vii. 17] Again,t since E by multiplying F, G has made C, B respectively, 30 therefore, as F is to G, so is C to B. [vn. 1 7] But, as C is to .ff, so is Z? to ^ ; therefore also, as D is to E, so is F to G. And alternately, as D is to F, so is E to G. [vn. 13] Therefore A, B are similar plane numbers; for their sides 3S are proportional. q. e. d. J J. For, since F 17. C to B. The tent has clearly suffered oomiplion here. It is not neccssaty to inftr from otfaer facts th^t, ^ 2^ is to £, so \& A Ui C\ for this is part of the hypotheses (II. 6, 7). Again, there is no enplanation of the statement {1. 55) that ^ by multiplying E has made C. It is the statement and explanation of this latter fact which are atene wanted ; after which the proof proceeds as in 1. 18. We might therefore luiHtitute for It. J5 — »8 Ibe following, "For, since £ measures C the same number of times that D measures A [1. 8], that is, according to the units in F [1. loj, therefore F by multiplying E has made C. And, since E by moltiplying F, G," etc. etc. This proposition is the converse of vi 11, 18. \i a,e, b are in geometrical pn^ession, a, b are " similar plane numbers." Let a : j3 be the ratio a i ( (and therefore also the ratio f : i) in its lowest tenns. Then [vn. 20] a ~ ma, c - mfi, where m is some integer, ' ' ' ' f = wo, i = »ft where « is some integer. VIII. 30, *i] PROPOSITIONS JO, a I m Thus a, ^ are both products of two factors, i.e. plane. ■ , , , ,. Again, a : fi = a : e = t : A ^ = m:n. ■<.,.. ,, [v". i8] Therefore, alternately, a:m='^:n, • [vii, 13] and hence ma, «j3 are similar plane numbers. [Our notation makes the second part still more obvious, for ^ =»(jS=«(x.] I Proposition 21. If two mean proportional numbers fali between ia)9 numbers, the numbers are similar solid numbers. For let two mean proportional numbers C, D fall between the two numbers A, B ; I say that A, B are similar solid numbers. Ill' A E- ^ ■ ■ •■• • B F •':■ ■' .. ^ c a .,/,'• D M- ,\i I, . , «— . K- O L- M For let three numbers E, F, G, the least of those which have the same ratio with A, C, D, be taken ; [vii. 33 or vin. a] therefore the extremes of them £, G are prime to one another. [vni. 3] Now, since one mean proportional number F has fallen between £, G, therefore E, G are similar plane numbers. [vm. aoj Let, then, H, K be the sides of E, and L, M of G, Therefore it is manifest from the theorem before this that £■, F, G are continuously proportional in the ratio oi H to L and that of K to M. Now, since £, F, G are the least of the numbers which have the same ratio with A, C, D, and the multitude of the numbers E, F, G Is equal to the mtiltitude of the numbers A, C, D, therefore, ex aequali, as E is to G, so is A to /?. [vu. 14] But B, G are prime, primes are also least, [vii. ai] and the least measure those which have the same ratio with jifS BOOK Viri ' "^ fviii. 31 them the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent ; [vn. 20] therefore £ measures A the same number of times that G measures £>. Now, as many times as E measures A, so many units let there be in TV. Therefore JV by multiplying £ has made A. But £ is the product of //, K ; therefore N by muhiplying the product of J/, K has made A, ,. Therefore A is solid, and H, K, N axe. its sides. ,^ . Again, since E, F, G are the least of the numbers which have the same ratio as C, D, B, therefore E measures C the same number of times that G measures B. Now, as many times as E measures C so many units let there be in O. Therefore G measures B according to the units in 0\ therefore O by muhiplying G has made B. But G is the product of L, M; therefore O by multiplying the product of L, M has made B, Therefore B is solid, and L. M, O are its sides ; therefore A, B are solid. I say that they are also similar. For since N, O by multiplying E have made A, C, therefore, as N is to O, so is A to C, that is, E to F. [vn. 18] But, as E is to F, so is H 10 L and K to M\ therefore also, as // is to Z, so is A" to ^ and N to O. And H, K, N are the sides of A, and O, L, M the sides of^. Therefore A, B are similar solid numbers. q, e. d. The converse of viu. 19. If a, e, d, b are in geometrical progression, a, b are "similar solid numbers." Let a, ;3, y be the least nunibers in the ratio of a, ^, d (and therefor* also of i-, rf, b). [vii. 33 or vui. »■ Therefore a, y are prime to one another, , .,.,., [vhi. 3^ They are also " similar plane numbers." ' [vni. 30 Let a. = mn, y =pf, where m:n-p:q. ■.-■■-. viii. ii-ij] PROPOSITIONS zi—i3 j79 Then, by the proof of viii, 20, a :j3 = m:/ = « : ^. Now, ex aegua/i, a ; J-a : y, [vil. 1 4} and, since a, y are prime to one another, B = m, d-ry, where r is an integer. But a = mn\ therefore a = rw«, and therefore a is " solid." Again, ex aequaii, c : &- a. : y, and therefore c = ja, b = sy, where J is an integer. Thus b = spg, and d is therefore " solid." Now a ; j3 = a : e = ra: so. = r:s. [vii. 18] And, from above, a: fi-m : p- n : g. Therefore r \ s = mip = » -.g, and hence a, b are similar sohd numbers. Proposition 22. /^ ^A^^e numbers be in continued proportion, and the first be sqxtare, the third will also be square. Let A, Bt C" be three numbers in continued proportion, and let A the first be square ; I say that C the third is also square. "' For, since between A, C there is one mean proportional number, B, therefore A, C are similar plane numbers. [viii. 30] But A is square ; " ' ' therefore C is also square. Q. e. d. A mere application of viu. zo to the particular case where one of the " similar plane numbers " is square. Proposition 23. I/four numbers be in continued proportion, and the first be cube, the fourth will also be cube. Let A, B, C, i? be four numbers in continued proportion, and let A be cube ; I say that D is also cube. * For, since between A, D there c are two mean proporti onalnumbers q B, C, therefore A, D are similar solid numbers, [vui. ai] |9q n- BOOK VIII >i'l [vui. 33—35 But^ is cube; 1 .,k ..• r <■ therefore Z? is also cube. ' • Q, E, D. A mere application of viii. a i to the case where one of the " similar solid numbers " is a cube. _, , j ., ,. t Proposition 34. ,. „jj // two numbers have to one another the ratio which a square number has to a square number, and the first be square, the second wiU also be square. For let the two numbers A, B have to one another the ratio which the square number C has to the square number D, and let j4 be A square ; B I say that ^ is also square. q For, since C, D are square, C, D are similar plane numbers. Therefore one mean proportional number falls between C, D. [viii. 18] And, as C is to Z?, so is .(4 to 5 ; therefore one mean proportional number falls between A, B also. [viii. 8] And j4 is square ; .■i'- therefore B is also square. ; , . [vm, 21] Q. E. D. If « ; # = ^' ; (f", and « is a square, then b is also a square. For f', dy have one mean proportionai (d. [vm. 18] TheKfoie a, b, which are in the same ratio, have one mean proportional. [vm. 8] And, since a is square, b must also be a square. , . [vm, 33] Proposition 25. If two numbers have to one another the ratio which a cube number has to a cube number, and the first be cube, the second will also be cube. For let the two numbers A, B have to one another the ratio which the cube number C has to the cube number D, and let A be cube ; 1 say that B is also cube. ' ^ • • 1 >ii «ii. »5, a6] PROPOSITIONS 13— »fi 3«i For, since C, D are cube, C, /? are similar solid numbers. Therefore two mean proportional numbers fall between D. [viii. ig] C, D A E- B F- O ■ And, as many numbers as fall between C, D in continued proportion, so many will also fall between those which have the same ratio with them ; [vm. 8] so that two mean proportional numbers fall between A, B also. Let £, Fso fall. Since, then, the four numbers A, E, F, B are in continued proportion, and A is cube, therefore B is also cube. [vm. aj] Q. E. D, ir d : b = ('id', and a is a cube, then b is also a cube. For c', d* have two mean proportionals, [viil. 19] Therefore a, b also have two mean proportionals, [vin. 8] And a is a cube : therefore £ is a cube. [vm. 33] Proposition 26. Similar plane numbers have to one another the ratio which a square numier has to a square number. , .. Let A, B be similar plane numbers ; I say that A has to B the ratio which a square number has to a square number. l» .1 « c- D e- For, since A, B are similar plane numbers, therefore one mean proportional number falls between A, B, [vm. 18] ^ '^- BOOK VIII [viii. a6, 37 Let it so fall, and let it be C; —- ^- ' - and let /}, £, F, the least numbers of those which have the same ratio with A, C, B, be taken ; [vu, 33 or vm. 2] therefore the extremes of them D, F are square, fvm. i. Pot.] And since, as D is to F, so is A to B, and D, F are square, therefore A has to B the ratio which a square number has to a square number. Q. E. D. If a, ^ are similar "plane numbers," let ^ be the mean proportional between them, [vui. 18 Take a, P,y the smallest numbers in the ratio of a, c, t. [vti. 33 or viii. 2 Then a, y are squares. [vii[, j, Por.^ Therefore a, i are in the ratio of a square to a square. PrOI'OSITION 27. Similar solid numbers have to one another the ratio which a cube number has to a cube number. Let A, Bh^ similar solid numbers ; . . I say that A has to B the ratio which a cube number has to a cube number. A c- B D- E — F Q H For, since A, B are similar solid numbers, therefore two mean proportional numbers fall between A, B. [viii. 19] Let C, D so fall, and let E, F, G, H, the least numbers of those which have the same ratio with A, C, D, B, and equal with them in multitude, be taken ; [vii. 33 or vm, j] therefore the extremes of them E, H are cube. [vm. », Por.] And, as E\iXo H,^o\% A Xa B\ therefore A also has to B the ratio which a cube number has to a cube number. ,^ Q. E. D. vui. 37] PROPOSITIONS 26, 27 383 The sime thing as viii. 26 with cubes. It is proved in the same way except that viii. 19 is used instead of viii. 18, The last note of an-Nairlzi in which the name of Heron is mentioned is on this proposition. Heron is there stated (p. 194 — 5, ed. Curtze) to have added the two propositions that, I. If two numbtri havt to ons another the ratio 0/ a square to a square, the numbers are similar f lane numbers ; 1, If two numbers have to one another the ratio of a cube to a cube, the numbers are similar solid numbers. The propositions are of course the converses of viii. a 6, 27 respectively. They are easily proved (i) If a:b = c':<P, then, since theie is one mean proportional {ed) between c\ d*, [viii. 1 1 or 18] there is also orx mean proportional between a, b. [viii. 8] Therefore a, b are simitar plane numbers. [viti. 20] (1) is similarly proved by the use of viii. 12 or it^, viii. 8, vtit. 31. The insertion by Heron of the first of the two propositions, the converse of vm. a6, is fwrhaps an argument in favour of the correctness of the text of IX. 10, though (as remarked in the -note on that proposition) it does not give the easiest proof Cf Heron's extension ni vii. 3 tacitly assumed by Euclid in vii, 33. '.' lifi ". .'■\ Tfpf.-i ?Rfi ';•• .. -. ..; 'j.'i ' I, ..'. , 1 ' 'i .'I I,' V'-. ,.i I , »; ■ i •' .; I*}' * . ,1 •.<K..vJ •.! ' iTi-iL;ii.t(i la«" .r-'" -1 I'/'j.-' ' .{ <l-;-iW'-, ' ■ V. ,.'•■•('.1 I; JLl'i -■rajrt a M- BOOK IX. A- B- C- Proposition r. // two similar plane numbers by muUiplying one another make some number, the product will be square. Let A, B be two similar plane numbers, and let A by multiplying B make C; I say that C is square. For let A by multiplying itself make D. ^ Therefore /? is square. Since then A by multiplying itself has made D, and by multiplying B has made C, therefore, as ^ is to ^, so is Z? to C [vti. 17] And, since A, B are similar plane numbers, therefore one mean proportional number falls between A, B. [vm. 18] But, if numbers fall between two numbers in continued proportion, as many as fall between thetn, so many also fall between those which have the same ratio ; [vni. 8] so that one mean proportional number falls between D, Calso. And D is square ; therefore C is also square. [vm. la] Q. E. D. The product of two similar pkne numbers b a square. Let a, ^ be two similar plane numbers. Now « : i = fl" ; oi. [vii. 1 7" And between a, b there b one mean profwrtional. [vm, \i Therefore between «' : «# there is one mean proportional. [vm, 8' And <^ is square ; therefore ab is square. [vm. ai] IX. *, 3l PROPOSITIONS 1—3 ^ ■ ■ . . : - ,"1 Proposition 2, - . .. 1 If two numbers by multiplying one another make a square number, they are similar plane numbers. Let A, B be two numbers, and let A by multiplying B make the square number C\ , . f I ' **!' ('■ est* ' I say that A, B are similar plane '^' ' numbers. b For let A by multiplying itself ^ make D ; ° therefore D is square. '^ " ' ' "' ■'"''>• Now, since ^ by multiplying itself has made D, and by multiplying B has made C, therefore, as ^ is to ^, so is Z) to C ' ^ '' '"■'-• [vn. 17] And, since D is square, and C is so also, therefore D, C are similar plane numbers. Therefore one mean proportional number falls between A C. [vin. 18] And, as D is to C so is .^ to ^ ; therefore one mean proportional number falls between v4,./? also. [viii. 8] But, if one mean proportional number fall between two numbers, they are similar plane numbers ; [vim. 20] therefore A, B are similar plane numbers. Q. E. D, ir oj is a square number, «, b are similar plane numbers, (The converse of IX. i<) For a ; b - a* : al^. [vn. ij] And a', ab being square numbers, and therefore similar plane numbers, they have one mean proportional. [vin. 18] Therefore a, b also have one mean proportional. [vni. 8] whence a. b are similar plane numbers. rvni. aol "" Proposition 3. If a cube number by multiplying itself make some number, the product vnll be cube. For let the cube number A by multiplying itself make B ; I say that B is cube. 386 BOOK IX [ix. 3 For let C, the side of ^, be taken, and let C by multiplying itself make D. It is then manifest that Cby muhiplying a D has made A. ^ Now, since C by multiplying itself has c- d — made D, therefore C measures D according to the units in itself. But further the unit also measures C according to the units in it; therefore, as the unit is to C, so is C to D. [vn, Def. ao] Again, since C by multiplying D has made A^ therefore D measures A according to the units in C. But the unit also measures C according to the units in it ; therefore, as the unit is to C so is Z? to -(4. But, as the unit is to C, so is C to Z? ; therefore also, as the unit is to C, so is C to D, and D to A. Therefore between the unit and the number A two mean proportional numbers C, D have fallen in continued proportion. Again, since A by multiplying itself has made B, therefore A measures B according to the units in itself. But the unit also measures A according to the units in it; therefore, as the unit \% \.<a A , ?^ '\% A to B. [vu. Def. 20] But between the unit and A two mean proportional numbers have fallen ; therefore two mean proportional numbers will also fall between A, B. {vtu. 8] But, if two mean proportional numbers fall between two numbers, and the first be cube, the second will also be cube. [vni. 23] And A is cube ; therefore B is also cube. ' Q. e. d. The product of t^ into itself, or a" , a', is a cube. For I : rt = a : fl' = a* : b'. Therefore between i and a' there aie two mean proportionals. Also I : a^ = u' : a* . o*. Therefore two mean proportionals fall between a' and «' . a*. [vui, 8] (U is true that viii. 8 is only enunciated of two pairs of numbers, but the proof is equally valid if one number of one pair is unity.) And a* is a cube number: therefore o^ . <t' is also cube. [viii. 13] IX. 4, 5] HROPOSITIONS 3—5 38J Proposition 4. If a cuie number by multiplying a cube number make some number, the product will be cube. For let the cube number A by multiplying the cube number B make C ; I say that C is cube. A For let A by multiplying^ e— — itself make D ; c therefore D is cube. [ix. 3] ° And, since A by muUtply- ing itself has made D, and by multiplying B has made C therefore, as A is to B, so is D to C [vii. ij] And, since A, B are cube numbers, A, B are similar solid numbers. Therefore two mean proportional numbers fall between A, B ; [viii. 19] so that two mean proportional numbers will fall between D, C also. [viii. 8] And D is cjbe; therefore C is also cube [vm. 13] Q. E. D. The product of two cubes, say «* . ^, is a cabe. For a' : *' = a'.o' : d*.*"." [vii. 17] And two mean proportionals fall between a*, f which are similar solid numbers, [vili. 19' Therefore two mean proportionals fall between 4' .i^, ^.y [viii. 8' B jt a* , o* is a cube : [ix. 3 therefore a*. ^ is a cube. ' ' [vili, 23 Proposition 5. // a cube number by multiplying^ any number make a cube number, Ike multiplied number will also be cube. For let the cube number A by multiplying any number B make the cube number C; I say that B is cube. /^ For let A by multiplying s « .<•' itself make D ; c therefore D is cube. [ix. 3] P^ ^ BOOK IX [IX, 5, 6 Now, since ^ by multiplying itself has made J?, and by- multiplying £ has made C, therefore, as >4 is to B, so is Z? to C. v^<fv,a ■:■'.*' [vn. 17] And since Z>, C are cube, ' •"'**'i»^, ►» they are similar solid numbers. Therefore two mean proportional numbers fall between D, C [vill, ig] And, as Z* is to C, so is ^ to ,5 ; therefore two mean proportional numbers fall between A, B also, [vin, 8] And A is cube ; therefore B is also cube, [viii. aj] If the product cfb is a cube number, b is cube. " ' By IX. 3, the product a',<i' is a cube. . V. ■ ,. • And tf , <^ : tfb = tf i b. [vn. 17] The first two terms are cubes, and therefore "similar sohds"; therefore there are two mean proportionals between them. [^i"- 19] Therefore there are two mean proportionals between a*, b. [vn:. 8J And 0* is a cube : therefore i is a cube number. [viiL 93] Proposition 6, If a nttmber by muUiplytng itself make a cube number, it will itself also be cube. For let the number A by multiplying itself make the cube number B ; I say that A is also cube, ■ '■""•'"""•I-". a . For let A by multiplying B make C. ^ Since, then, A by multiplying itself o has made B, and by multiplying B has made C, therefore C is cube. -.^- ■.u.av. -. -/w .. .< And, since A by multiplying itself has made B, therefore A measures B according to the units in itself. But the unit also measures A according to the units in it Therefore, as the unit is to A, so is A to B, [vn. Def. ao] And, since A by multiplying B has made C, therefore B measures C according to the units in A. But the unit also measures A according to the units in it. IX. 6, 7] PROPOSITIONS s— 7 Therefore, as the unit is to A, so is B to C. But, as the unit is to A, sols A to B ; therefore also, as ^ is to B, so is ^ to C. . And, since B, C are cube, they are similar solid numbers. Therefore there are two mean between B, C. And, as B is to C, so is ^^ to B. Therefore there are two mean between A, B also. And B is cube ; therefore A is also cube. 3«9 [vii. Def. so] proportional proportional numbers [viii. 19] numbers [vm. 8] [cf. vm. 23] Q. E. D, If a" IS a cube number, a is also a cube. For I ; a = fl : a' = a' : a*. Now a\ (^ are both cubes, and therefore "similar solids "; therefore ther« :\je two mean proportionals between them. [vm. 19] Therefore there are two mean proportionals between a, a', [vm. 8] And a* is a cube : therefore a is also a cube number. [vm. 23] It will be noticed that the last step is not an exact quotation of the result of vtn. «3, because it is there the first of four terms which is known to be a cube, and the last which is proved to be a cube ; here the case is reversed. But there is no difficulty. Without inverting the proportions, we have only to refer to vm. 2r which proves that a, cf, having two mean proportionals between them, are two similar solid numbers ; whence, since a" is a cube, a is also a. cube. Proposition 7. Tf a composite number by multiplying any number make some number, the product will be solid. For let the composite number A by multiplying any number B make C\ f^, I say that C is solid. ^ For, since^ is composite, q it will be measured by some ^ ^ number. [vn. Def. 13] Let it be measured hy D\ and, as many times as D measures A, so many units let there be in B. |9l9 BOOK IX [ix. 7, S [o Since then D measures A according to the units in £", therefore E by multiplying D has made A. \ym Def. 15] And, since A by multiplying B has made C, and A is the product of D, E, therefore the product of D, E by multiplying B has made C. Therefore C is solid, and D, E, B are its sides. Q. E. D. 'X^ Since a composite number is the product of two factors, the result of multiplying tt by another number is to produce a ~n umber which is the product of three factors, i.e. a "solid number." ; . ..I . . -. , . -.^ ,...-:), , , , Proposition ST. If as many numbers as we please beginning from an unit, be in continued proportion, the third from the unit will be square, as will also those which successively leave out one ; the fourth will be cube, as will also all those which leave out two; and the seventh will be at once cube and square, as will also those which leave out five. Let there be as many numbers as we please, A, B, C, D, B, F, beginning from an unit and in con- tinued proportion ; A I say that B, the third from the unit, is ^ square, as are also all those which leave p out one ; C, the fourth, is cube, as are g also all those which leave out two ; and p E, the seventh, is at once cube and square, as are also all those which leave out five. For since, as the unit \s to A , so is A to B, therefore the unit measures the number A the same number of times that A measures. .5, [vu. Def 26] But the unit measures the number A according to the units in it ; therefore A also measures B according to the units in A, Therefore A by multiplying itself has made B; therefore B is square. 1 . .• And, since B, C, Z? are in continued proportion, and B is square, therefore D is also square. [vtit. a 3] IX. 8] PROPOSITIONS 7, 8 39* For the same reason v. , t .1. .u m.- r IS also square. Similarly we can prove that all those which leave out one are square. I say next that C, the fourth from the unit, is cube, as are also all those which leave out two. For since, as the unit is to W, so is 3 to C, therefore the unit measures the number A the same number of times that B measures C. But the unit measures the number A according to the units in A ; therefore B also measures C according to the units in A. Therefore A by multiplying B has made C. Since then A by multiplying itself has made B, and by multiplying B has made C, r I ■ • I'l ■ .^ I therefore C is cube. And, since C, D, E, F are in continued proportion, and C is cube, •_■(■.■ 11'*" 'J"^! • '■':! ' therefore F is also cube. fvui. 33] But it was also proved square ; therefore the seventh from the unit is both cube and square. Similarly we can prove that all the numbers which leave out five are also both cube and square. Q. E. D. If t, a, og, o,, ... be a geometrical progression, then a,, a^, a„ ... are ^uaies; a,, a,, a,, ... are cubes ; tf„ a„, ... are both squares and cubes. ■'"■ '* ' ' ' '"' Since 1 : a=^a : at, a, = a". And. since a,, a,, a^ are in geometrical progression aitd a, (= a*) is a square^ a, is a square. [viii. 31] Similarly a,, a^, ... are squares. Next, I : a = o, ; uj ., ..•; whence aj=a', a cube number. "'^ , ' ~" And, since Oj, anOi, a, are in geometrical progression, and a, is a cube, a, is a cube. [viii. 33] 39» BOOK IX [tx. 8, 9 Similarly a„ a,,, ... are cubes. ' Vi t Clearly then a,, Ou, o„, ... are both squares and cubes. The whole result is of course obvious if the geometrical progression is written, with our notation, as I, a, a', a", a*, ... e". Proposition 9. T/as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be square, all the rest will also be square. And if Ihs number after the unit be cube, all the rest will also be cube. Let there be as many numbers as we please, A, B, C, D, E, F, beginning from an unit and in con- tinued proportion, and let A, the number a after the unit, be square ; ^ I say that all the rest will also be square. ^ Now it has been proved that B, the £ third from the unit, is square, as are also f all those which leave out one ; [ix. 8] I say that all the rest are also square. For, since A, B, C are in continued proportion, and A is square, , therefore C is also square. [vm. n] Again, since B, C, D are in continued proportion, and B is square, D is also square. [vm. 22] Similarly we can prove that all the rest are also square. Next, let A be cube ; I say that all the rest are also cube. Now it has been proved that C, the fourth from the unit, is cube, as also are all those which leave out two ; [ix, 8] I say that all the rest are also cube. For, since, as the unit is to y^, so is /^ to B, therefore the unit measures A the same number nf times as A measures B. But the unit measures A according to the units in it ; therefore A also measures B according to the units in itself; therefore A by multiplying itself has made B, IX. 9, lo] PROPOSITIONS 8—10 393 And ^ is cube. - ' But, if a cube number by multiplying itself make some number, the product is cube. [ix. 3] Therefore B is also cube. And, since the four numbers A, B, C, D are in continued proportion, and >^ is cube, . ., . - , ,^ :.. ... D also is cube. , , ■ mii [v'm. 23] For the same reason E is also cube, and similarly all the rest are cube. ., Q. E. D. If I, a*, tf,, a,, a^, ... are in geometrical progression, t>,, a,, a^, ... are all squares; and, if I, a", di, Ott ^t ■•• ^c in geomt^Crical progression, o^, a,, ... are all cubes, (i) By IX. 8, IT,, 1I4, a,, ... are all squares. And, a', ii„ a, being in geometrical progression, and o' being a square, a^'viA square. [viii, 23] For the same reason a,, Ot, ... are at) squares. (1) By IX. 8, 17,, a,, a„ ... are all cubes. . Now I : o* = «* : «j. Therefore <»i = (J* . o*, which is a cube, by ix. 3. And, <^, 011 "■> <it being in geometrical progression, and a* being cube, a, is cube. [^"i- '3] Similarly we prove that o, is cube, and so on. The results are of course obvious in our notation, the series being (i) I, a', a*, a*. ... o* Proposition id. l/as many numbers as we piease beginning from an unit be in continued proportion, and the number after the unit be not square, neither will any other be square except the third from the unit and all those which leave out one. And, if the number after the unit be not cube, neither will any other be cube except the fourth from the unit and all those which leave out two. Let there be as many numbers as we please. A, B, C, D, E, F, beginning from an unit and in continued proportion, and let A^ the number after the unit, not be square ; 3m BOOK IX [ix. lo I say that neither will any other be square except the third from the unit <and those which leave out one > . a ' For, if possible, let C be square. b But B is also square ; [«. 8] 2 [therefore B, C have to one another ° the ratio which a square number ^ has to a square number]. ^ And, as B is to C, so is ^ to ^ ; ' "'*" • therefore A, B have to one another the ratio which a square number has to a square number ; [so that Ay B are similar plane numbers]. [vin. 26, converse] And B is square ; therefore A is also square : which is contrary to the hypothesis. Therefore C is not square. Similarly we can prove that neither is any other of the numbers square except the third from the unit and those which leave out one Next, let A not be cube, 1 say that neither will any other be cube except the fourth from the unit and those which leave out two. For, if possible, let D be cube. Now C is also cube ; for it is fourth from the unit. [ix. 8] And, as C is to /), so is ^ to C ; therefore B also has to C the ratio which a cube has to a cube. And C is cube ; therefore B is also cube. [vm. 15] And since, as the unit is to A, so is A to B, and the unit measures A according to the units in it, therefore A also measures B according to the units in itself ; therefore A by multiplying itself has made the cube number ^, But, if a number by multiplying itself make a cube number, it is also itself cube. [ix. 6] Therefore A is also cube : which is contrary to the hypothesis. Therefore D is not cube. IX. lo, itj PROPOSITIONS lo, II 395 Similarly we can prove that neither is any other of the numbers cube except the fourth from the unit and those which leave out two. Q. E. D. ir I, a, a^, a,, at, ... be a geometrical progression, then (i), if a is nut a square, none of the terms will be square except a^, Of, n,>, ...; and (a), if a is not a cube, none of the terms will be cube except «j, a„ a„, With reference to the first part of the proof, viz. that which proves that, if a, is a square, a must be a square, Heiberg remarks that the words which I have bracketed are perhaps spurious; for it is easier to use vui. 24 than the converst of viii, 26, and a use of vui. 24 would correspond better to the use of VIII. 15 in the second part relating to cubes. I agree in this view and have bracketed the words accordingly. {See however note, p, 383, on converses of viii, 26, 27 given by Heron.) If this change be made, the proof runs as follows. (i) If possible, let 03 be square. ' ' Now Q;, ; dj = a : fij. But ^, is a square. [ix. 8] Therefore a is to a, in the ratio of a square to a square. And <(j is square ; therefore a is square [viii. 24] : which is impossible. (3) If possible, let «< be a cube. Now a, ; a, = a, : a,. And a, is a cube. [ix. 8] Therefore a, is to «, in the ratio of a cube to a cube. And a, is a cube : therefore a, is a cube. [viu. 35] But, since i : a = a •.oj, tij = a'. And, since a* is a cube, a must be a cube [ix. 6] : which is impossible. The propositions viii. 24, 35 are here not quoted in their exact form in (hat the _firsi and uccnd squares, or cubes, change places. But there is no difficulty, since the method by which the theorems are proved shows that either inference is equally correct Proposition ii. If as many numbers as we please beginning from^ an unit be in continued proportion, the less measures the greater according to some one of the numbers which have place among the propor- tional numbers. 30 , BOOK IX [ix. ti •In Let there be as many numbers as we please, B, C, D, E, beginning from the unit A and in con- tinued proportion ; ^ I say that B, the least of the numbers B, a C, D, E, measures E according to some c- one of the numbers C, D. o For since, as the unit A is to B, so ^ is D to E, therefore the unit A measures the number B the same number of times as D measures E \ ^ V'i"., . It. T , therefore, alternately, the unit A measures D the same number of times as B measures E. [vii. 15] But the unit A measures D according to the units in it ; therefore B also measures E according to the units in D ; so that B the less measures E the greater according to some number of those which have place among the proportional numbers. — PoRiSM. And it is manifest that, whatever place the measuring number has, reckoned from the unit, the same place also has the number according to which it measures, reckoned from the number measured, in the direction of the number before it. — Q. E. D. The proposition and the porism together assert that, if t, a, a,, ... a„ be a geometrical progression, a, measures a» and gives the quotient o,., {r < it). Eudid only proves that a^ = a.a^.„ as follows. Therefore i measures a the same number of times as «„., measures a,. Hence i measures a„., the same number of times as a measures a, ; . . [vn. 15} that IS, a, = a,ai,_,. We can supply the proof of the porism as follovrs. I : a = Br : a,^,, (J : n, = «,+! : a^+j, whence, tx aequaii, I : a,^, = fl, : a,. [vil. 14] It follows, by the same argument as before, that With our notation, we have the theorem of indices that IX. i2l PROPOSITIONS II, I* ^ Proposition 12. If as many numbers as we please beginning from an unit be tn conlitmed proportion, by however many prime numbers (he lasi is measured, the next to the unit will also be measured by ike same. Let there be as many numbers as we please, A, B, C, D, beginning from an unit, and in continued proportion ; I say that, by however many prime numbers D is measured, A will also be measured by the same. A P- " ,. ', B Q • O For let /? be measured by any prime number £ ; I say that £ measures A. I" For suppose it does not; " - ' now £ is prime, and any prime number is prime to any which it does not measure ; [vn. 29] therefore £, A are prime to one another. And, since E measures D, let it measure it according to F, therefore £ by multiplying F has made D. Again, since A measures D according to the units in C, [ix, 1 1 and For,] therefore A by multiplying C has made D. But, further, £ has also by multiplying F made D ; therefore the product of ^, C is equal to the product of £, F. Therefore, as A is to B, so'is F to C. [vii> 19] But .(4, .£ are prime, .< j :"i : > primes are also least, [vn. 21] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent ; ., ^ [vn. 20] therefore E measures C. ., Let it measure it according to G; - ' • therefore £ by multiplying G has made C. But, further, by the theorem before this, A has also by multiplying B made C. [ix. 1 1 and Por.] ^ c> . BOOK IX- [ix. 13 Therefore the product of A, B is equal to the product of Therefore, as -,4 is to £", so is ff to J?. ''^ " [vil 19] But A, E are prime, primes are also least, [vii. ai] and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent : L^ii. 10] therefore H measures B. Let it measure it accordinp; \o H \ therefore E by multiplying H has made B, But further A has also by multiplyiner itself made B ; [.X.81 therefore the product of E, H is equal to the square on A. Therefore, as £" is to ^, so is ^ to //". [vii. 19] But A, E are prime, ..; •> I primes are also least, [vn. zi] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent ; [vn. so] therefore E measures A, as antecedent antecedent But, again, it also does not measure it : , which is impossible^ -iii.. • '...if! . /', Therefore E, A are not prime to one another. ^ ^ Therefore they are composite to one another. '' ''' " ' But numbers composite to one another are measured by some number, \^i\. Def. 14] And, since E ij by hypothesis prime, and the prime is not measured by any number other than itself, therefore E measures A, E, so that E measures A. > • ^ . . ' ,• [But it also measures /? ; ■"*-' ■ . i 1. iv- therefore E measures A, Z?.] Similarly we can prove that, by however many prime numbers D is measured, A will also be measured by the same. Q. E. D, If I, a, 0,, ... a^hea. geometrical progression, and a. be measured by any prime number >, a will a.tso be measured by /. IX. I*, 13] PROPOSITIONS 12, 13 399 For, if possible, suppose that p does not measure a J then,/ being prime, /, a are prime to one another. . ^ [vii. 29] Suppose a^^m.p. ,.".,. .,., Now • o, = a.a,.,. ' ' [ix. li] Therefore o . o,., = « . /, and a : p = m : a^-i. • ' > [vil. 19] Hence, n,/ being prime to one another, p measures ii„_i. [vil. zo, ir] By a reptetition of the same process, we can prove that p measures a„.j and therefore o,_„ and so on, and finally that p measures a. But, by hypothesis, / does not measure a : which is impossible. Hence p, a are not prime to one another : therefore they have some common factor. [vu. Def. 14] Butp is the only number which measures p; therefore/ measures a. Heiberg remarks that, as, in the iii0iirK, Euclid sets himself to prove that E measures j4, the words bracketed above are unnecessary and therefore perhaps interpoiatedi ■ » . • Proposition 13, //as many numbers as we piease beginning from an unit be in continued proportion, and the number after the unit be prime, the greatest will not be measured by any except those which, have a place among the proportional numbers. Let there be as many numbers as we please. A, B, C, D, beginning from an unit and in continued proportion, and let A, the number after the unit, be prime ; I say that Z?, the greatest of them, will not be measured by any other number except A, B, C. K e '••'•'^■^ B F ■ i' ' "■^■•\ C Q. O H For, if possible, let it be measured by E, and let E nqt be the same with any of the numbers A, B, C. It is then manifest that E is not prime. ' ' "' For, if £ is prime and measures D, it will also measure A [ix. 12], which is prime, though it is not the same with it : which is impossible. -rvrii .- 1. • . '. 4m BOOK. IX [ix. 13 Therefore -£" is not prime. ■-:■-■ .m.^. ■„.. . .1. Therefore it is composite. But any composite number is measured by some prime number ; [vii. 31] therefore B is measured by some prime number. I say next that it will not be measured by any other prime except A. For, if £ is measured by another, •■ . . and £ measures D, • , 1, that other will also measure D\ so that it will also measure A [ix. n], which is prime, though it is not the same with it : which is impossible. Therefore A measures E. And, since E measures D, let it measure it according to F. I say that F is not the same with any of the numbers A, B, C. For, if F is the same with one of the numbers A, B,C, and measures D according to A, therefore one of the numbers A,B,C also measures D according to E. But one of the numbers A, B, C measures D according to some one of the numbers A, B, C; [ix. 11] therefore E is also the same with one of the numbers A, B,C: which is contrary to the hypothesis. Therefore F is not the same as any one of the numbers A. B, C. Similarly we can prove that F is measured by A, by proving again that Fi& not prime. For, if it is, and measures Z?, it will also measure A [ix, i *], which is prime, though It is not the same with it : which is impossible ; therefore F is not prime. Therefore it is composite. But any composite number is measured by some prime number ; [va. 31 j therefore F is measured by some prime number. IX. 13] PROPOSITION 13 401 I say next that it will not be measured by any other prime except A. For, if any other prime number measures F, and F measures D, that other will also measure D ; so that it will also measure A [ix. u], which I's prime, though it is not the same with it : which is impossible. , .,^,, ,,u . Therefore A measures F. And, since E measures D according to ^^ ., .^^ therefore E by multiplying F has made D. ■ '• ' But, further, A has also by multiplying C made D; [ix, n] therefore the product of A, C is equal to the product of F, F. Therefore, proportionally, as A is to E, so is F to C. [vii. 19] But A measures E ; . therefore F also measures C, Let it measure it according to G. "' ' Similarly, then, we can prove that G is not the same with any of the numbers A, B, and that it is measured by A. And, since F measures C according to G therefore F by multiplying G has made C. But, further, A has also by multiplying B made C \ [ix. it] therefore the product of A, B is equal to the product of F, G. Therefore, proportionally, as A is to F, so is G to B. [v"' '9] But y4 measures /^; " ' ' ■. ', ,, therefore G^ also measures ^. ' •"•' Let it measure it according to H. Similarly then we can prove that H is not the same with A. And, since G measures B according to H, therefore G by multiplying H has made B, But further A has also by multiplying itself made B ; [ix. 8] therefore the product of H, G is equal to the square on A. Therefore, as // is to ^, so is v4 to 6^. [vil 19] jfjaa BOOK IX [ix. 13, 14 But A measures (7; ' ' ■ > ■ . ■ . ■,. 1 therefore H also measures A, which is prime, though it is not the same with it : which is absurd. Therefore D the greatest will not be measured by any other number except A, B, C. • Q. E. D. If I, a, a,, ... a„ b« a geometrical progression, and if a is prime, a. will not be measured by any numbers except the preceding terms of the series. ir possible, let a^ be measured by by a number different from all the preceding terms. Now b cannot be prime, for, if it were, it would measure a. [ix. 12] Therefore b is composite, and hence will be measured by somt prime number [vii. 31], say p. Thus p must measure a„ and therefore a [ix. i i] ; so that / cannot be different from a, and b is not measured by any prime number except a. Suppose that a^~b . r. Now f cannot be identical with any of the terms a, a,, ... a^-ti for, if it were, 6 would bt identical with another of them: [ix, 11] which is contrary to the hyfwthesis. We car non- prove (just as for b) that c cannot be prime and cannot be measured by any prime number except a. Since l>.e-a^-a . a^.i, [ix. Ii] a:i = e: a,_, , whence, since a measures i, IT measures a^-i. Let a^_, = t .d. We now prove in the same way that d is not identical with any of the terms 0, a■^, ... a,_i, is not prime, and is not measured by any prime except a, and also that d measures a,.|. Proceeding in this way, we get a last factor, say k, which measures a though different from it : which is absurd, since a is prime. Thus the original supposition that a„ can be measured by a number S different from all the terms a, a^, ... a„_, must be incorrect. Therefore etc. Proposition 14. //a number be (he least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it. For let the number A be the least that is measured by the prime numbers B, C, D; IX. 14] PROPOSITIONS 13, 14 403 I say that A v/iW not be measured by any other prime number except B, C, Z>. For, if possible, let it be measured by the prime number £, and let £ not be the same with any one of the numbers B. C, D. A B F Now, since E measures A, let It measure it according to F\ therefore E by multiplying F has made A. And A is measured by the prime numbers B, C, D. But, if two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers ; [vii. 30] therefore B, C, D will measure one of the numbers E, F. Now they will not measure E ; for E is prime and not the same with any one of the numbers B, C D. Therefore they will measure F, which is less than A : which is impossible, for A is by hypothesis the least number measured by B, C, D. Therefore no prime number will measure A except B, C, D. Q. E. D. In other words, a number can be resolved into prime factors in only one way. Let a be the least number measured by each of the prime numbers i, t, d, ... k. If possible, suppose that a has a prime factor/ different from />, c, d, ... k. Let a-p.m. Now /i,(,d, ... a, measuring «, must measure one of the two factors/, m. [vii. 30] rhey do not, by hypothesis, measure p ; therefore they must measure iw, a number less than a: "•.)""•";.(-'. which is contrary to the hypothesis. Therefore a has no prime factors except i, (, d, ... k. 404 *» - BOOK IX I [ix. IS Proposition 15, // three numbers in continued proportion be the least of those which have the same ratio with them, any two whatever added together will he prime to the remaining number. Let A, B, C, three numbers in continued proportion, be the least of those which have the same ratio with them ; * b ■ I say that any two of the numbers c A, B, C whatever added together are 0— J — f prime to the remainingnumber, namely . A, B to C\ B, Cto A ; and further A, C to B. For let two numbers DB, EF, the least of those which have the same ratio with A, B, C, be taken. [vni. 1] It is then manifest that I>B by multiplying itself has made A, and by multiplying BB has made B, and, further. BB by multiplying itself has made C. [vin. a] Now, since DB, BBare least, they are prime to one another. [vii. «] But, if two numbers be prime to one another, their sum is also prime to each ; [vii. a8] therefore DB is also prime to each of the. numbers DB, BB. But further D£ is also prime to BB ; ^ therefore DB, DB are prime to BB. But, if two numbers be prime to any number, their product is also prime to the other ; [vii. 84] so that the product of BD, DB is prime to BF\ hence the product of FD, DB is also prime to the square on BF. [vii. »j] But the product of BD, DE is the square on DB together with the product of DB, BF; [n. 3] therefore the square on DB together with the product of DB, BF is prime to the square on BF. And the square on DB is A, the product of DE, BB is B, .,...., and the square on BF is C; therefore A, B added together are prime to C. IX. is] proposition 15 405 Similarly we can prove that B, C added together are prime to A. I say next that A, C added together are also prime to B. For, since DF is prime to each of the numbers DE, EF, the square on DF is also prime to the product of DE, EF. [vii. 24, js] But the squares on DE, EF together with twice the pro- duct of DE, EFa.K equal to the square on DF; [11. 4] therefore the squares on DE, EF together with twice the product of DE, EF are prime to the product of DE, EF. Separando, the squares on DE, EF together with once the product of DE, EF^e prime to the product of DE, EF. Therefore, separando again, the squares on DE, EF are prime to the product oi DE, EF. '' "'"'<• "- And the square on Z?^ is v4, :V . •.' . . the product of DE, EF is B, and the square on EF is C. : " Therefore A, C added together are prime to B. Q. E, D. If a, 6, c he a. geometrical progression in the least terms which have a given common ratio, (i + f), (c+ 3), (a + *) are respectively prime to a, d, e. Let a : ^ be the cominon ratio in its lowest terms, so that the geometrical progression is a\ a/9, f?. ' ' [VHI. »J Now, a, fi being prime to one another, ' ' a-*-/} is prime to both a and j8, " [vii. *8] Therefore (0 + ^)1 <> ^t* t^th prime to p. Hence (a + ^ a is prime to j3, [vit. 24] and therefore to ^; [vii. 25] Le- a' + afi is prime to ^, or ' a + i is prime to ^. " Similarly, ' ' c^ + jS* is prime to a', * * or * + f is prime to o. . ' ■' **''■ Lastly, a ■«■ ^ being prime to both a and j9, (a + ^)' is prime to a/9, [vii. 24, 25] or o' + /S* + Mj9 is prime to afi : whence a' + yS" is prime to aj8. The latter inference, made in two steps, may be proved by reducHo ad absurdum as Commandinus proves it If a* + ^ is not prime to o/S, let x measure them ; therefore x measures a* + ^ ■«- lafi as well as o^ ; hence a* + ^ ■«- 2 aj3 and aj3 are not prime to one another, which is contrary to the hypothesis. 4o6 BOOK IX [ix. i6, i? Proposition i6. If two numbers be prime to one another, the second will not be to any other number as the first is to the second. For let the two numbers A, Bhe. prime to one another ; I say that B is not to any other number as A is to B. A For, if possible, as A is to B, so let B be b to C. o Now A, B are prime, primes are also least, fvii. zi] and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent ; [vii. 20] therefore A measures B as antecedent antecedent. But it also measures itself; therefore A measures A, B which are prime to one another : which is absurd. Therefore B will not be to Q as A is to B. If a, h are prime to one another, they can have no integral third proportional. ^^ If possible, let a : 6 = i ; x. Therefore [vii. zo, 21] a measures i, and a. b have the commorv measure, d, which is contrary to the hypothesis. ^ ,.,.,,. a , . .• 'r-r Proposition 17. "» ■ - If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the last will not be to any other number as the first to the second. For let there be as many numbers as we please, A,B,C,D, in continued proportion, and let the extremes of them, A, D, be prime to one another ; ^ 1 say that D is not to any other ^ ^ number as A is to B, For, if possible, as A is to B, so let D he to £ ; therefore, alternately, as y^ is to Z?, so is .5 to .£", [vii. 13] : J-' - '^i -.■j '- ,^^ii '- '\ 13(1 til ' ■ • 1 I I'ur'- 1 • ?o; 1 \ti IX, 17, 18] PROPOSITIONS 16—18 407. But A, D are prime, ~ ' ' ' • 1 primes are also least, if.' i" i-* yj [vii.it] and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent. [vn. 20] Therefore A measures ^. And, as A is to B, so is B to C. Therefore B also measures C \ so that A also measures C. And since, as B is to C, so is C to D, and ^ measures C, therefore C also measures D. , , „ But y4 measured C ; so that A also measures D. , , But it also measures itself; therefore A measures A, D which are prime to one another : which is impossible. Therefore Z? will not be to any other number as A is to B. Ifd, 1(9, d„ ... Sh be a geometrical prc^iression, and a, a. are prime to one another, then a, a„ a„ can have no integral fourth proportional. For, if possible, let a ; a, = a, : jf, Therefore a : a^ = a^\ x, and hence [vn. 20, 21] a measures at. Therefore a, measures a,, [vii. Def. 20] and hence a measures a,, and therefore also ultimately a,. Thus a, a^ are both measured by a : which is contrary to the hypothesis. Proposition 18. Given two numbers, to investigate whether it is possible to find a third proportional to them. Let A, B be the given two numbers, and let it be required to investigate whether it is possible to find a third proportional to them. Now A, B are either prime to one another or not. And, if they are prime to one another, it has been proved that it is impossible to find a third proportional to them. [tx. 16] 4«8 >< BOOK IX <y.-' [ix. 18 Next, let A, B not be prime to one another, and let B by multiplying itself make C. Then A either measures C or does not measure it. A- B- First, let it measure it according to D ; therefore A by multiplying D has made C. But, further, B has also by multiplying itself made C ; therefore the product of ^, D is equal to the square on B, Therefore, as ^ is to B, so is /? to Z? ; [vii. 19] therefore a third proportional number D has been found to A,B. Next, let A not measure C ; I say that it is impossible to find a third proportional number to A, B. ^ For, if possible, let /?, such third proportional, have been found. Therefore the product of A, D is equal to the square on B. But the square on ^ is C; therefore the product of A, D is equal to C. Hence A by multiplying D has made C \ therefore A measures C according to D. But, by hypothesis, it also does not measure it : which is absurd. Therefore It is not possible to find a third proportional number xja A^ B when A does not measure C. q. e. d. Given two numbers a, 6, to find the condition that they may have an int^ral third proportional, (t) a, i must not be prime to one another. [ix. 16] (a) a must measure S'. For, if a, ^, f be in continued proportion. Therefore a measures ^. Condition (i) is included in condition (1) since, if l^ = ma, a and S cannot be prime to one another. The result is of course easily seen if the three terras in continued proportion be written "' "a' Hi}- IX. rg] PROPOSITIONS i8, 19 j|«9 Proposition 19. ^''i ■•'• " ' '■■■ ' Given three numben, to investigate when it is possible to find a fourth proportional to tkem. Let A, B,C\x:^ the given three numbers, and let it be required to investigate when it is ^^ _ possible to find a fourth proportional ^ , to them. g .,1' Now either they are not in con- p , , tinued proportion, and the extremes ^ of them are prime to one another ; or they are in continued proportion, and the extremes of them are not prime to one another ; or they are not in continued proportion, nor are the extremes of them prime to one another ; or they are in continued proportion, and the extremes of them are prime to one another. If then A, B, C are in continued proportion, and the extremes of them A, C are prime to one another, it has been proved that it is impossible to find a fourth pro- portional number to them. [ix. 17] tNext, let A, B, C not be in continued proportion, the extremes being again prime to one another ; I say that in this case also it is impossible to find a fourth proportional to them. For, if possible, let D have been found, so that, as y^ is to ^, so is C to D, and let it be contrived that, as ^ is to C, so is Z> to E. Now, since, as A is to B, so is C to D, and, as ^ is to C, so is Z? to .£", therefore, ex aequali, as ^4 is to C, so is C to B. [vii. 14] But A, C are prime, primes are also least, [vit. li] and the least numbers measure those which have the same ratio, the antecedent the antecedent and the consequent the consequent [vn, ao] Therefore A measures C as antecedent antecedent. 4W ^i BOOK IX ■• [ix. 19 But it also measures itself ; therefore A measures A, C which are prime to one another : which is impossible. Therefore it is not possible to find a fourth proportional to A. B, C.f Next, \et A, B, C be again in continued proportion, but let A, C not be prime to one another. I say that it is possible to find a fourth proportional to them. For let B by multiplying C make D ; therefore A either measures D or does not measure it. First, let it measure it according to E ; therefore A by multiplying E has made D. But, further, B has also by multiplying C made D ; therefore the product of A, E is equal to the product of B, C; therefore, proportionally, as A is to B, so is C to ^ ; [vii. 19] therefore E has been found a fourth proportional to A, B, C. Next, let A not measure D ; I say that it Is impossible to find a fourth proportional number to A, B, C. For, if possible, let E have been found ; therefore the product of A, E is equal to the product of^, C, [vii. 19] But the product if B, C is D ; therefore the product of A, E is also equal to D. Therefore A by multiplying E has made D ; therefore A measures D according to E, so that A measures D. But it also does not measure it : which is absurd. Therefore it is not possible to find a fourth proportional number to A, B, C when A does not measure D. Next, let A, B, C not he in continued proportion, nor the extremes prime to one another. And let B by multiplying C make D. . 1 • Similarly then it can be proved that, if A measures D, it is possible to find a fourth proportional to them, but, if it does not measure it, impossible, Q. e. d. IX. ig] PROPOSITION ig 411 Given three numbers a, b, e, to find the condition that they may have an integral fourth proportional. The Greek text of part of this proposition is hopelessly corrupt. Accord- ing to it Euclid takes four cases, (i) a,b,e not in continued proportion, and a, c prime to one another. (2) a, ^, c in continued proportion, and a, c not prime to one another. (3) o, *i <^ not in continued proportion, and a, t not prime to one another. (4) a,b,c in continued proportion, and a, c prime to one another. (4) is the case dealt with in ix. 17, where it is shown that on hypothesis {4) a fourth proportional cannot be found. The text now takes case (i) and asserts that a fourth proportional cannot be found in this case either. We have only to think of 4, 6, 9 in order to see that there is something wrong here. The supposed proof is also wrong. If possible, says the text, let rf Ije a fourth proportional to a, b, c, and lei e be taken such that b : c - d : e. Then, ex a^uali, n '. c-c ; e, whence a measures c : [vii, so, Ji] which is impossible, since o, c are prime to one another. But this does not prove that a fourth proportional d cannot be found ; it only proves that, if if is a fourth proportional, no integer e can be found to satisfy the equation , - .^ b : e-d ; e. Indeed it is obvious from ix. 16 that in the equation f- ■ •• . a : € = t \e " t cannot be integral. > . The cases {%) and (3) are correctly given, the first in full, and the other as a case to be proved "similarly" to it. These two cases really give all that is necessary. Let the product be be taken. Then, if a measures be, suppose bc = ad; therefore a : b = c : d, and ^ is a fourth proportional. But, if a does not measure be, no fourth proportional can be found. For, if X were a fourth proportional, ax would be equal to />c, and a would measure be. The sufficient condition in any case for the possibility of finding a fourth proportional to a, b, e is that a should measure be. Theon appears to have corrected the proof by leaving out the incorrect portion which I have included between daggers and the last case {3) dealt with in the last lines. Also, in accordance with this arrangement, he does not distinguish four cases at the beginning but only two. " Either A, B, C are in continued proportion and the extremes of them A, C are prime to one another; or not," Then, instead of introducing case (2) by the words "Next let A, B, C.to find a fourth proportional to them," immediately following the second dagger above, Theon merely says " But^ if not," [i.e. if it is not the case that a, b, ^ are in g.p. and a, ^ prime to one another] "let B by multiplying C make D," and so on. 4H BOOK IX ! [ix. 19, 10 August adopts Theon's fonn of the proof. Heibeig does not feel able to do this, in view of the superiority of the authority for the text as given above (P) ; he therefore retains the latter vrithout any attempt to emend it. '■ "•'"' Proposition 20. ' •'•""•'•' ^-" • 'M Prime numbers are more than any assigned muUitvde of prime numbers. iutt t ji .-i -irrw )>. Let ^, jff, C be the assigned prime numbers ; I say that there are more > i^' prime numbers than A, B, C. a— •''■ ' ' For let the least number ^ Q measured by ^, .5, C be c taken, e ^F and let it be DE ; let the unit i?/^ be added to ZJ.fi'. Then EF is either prime or not. ' '• First, let it be prime ; then the prime numbers A, B, C, EEhave been found which are more than A, B, C. Next, let ^"^ not be prime; ; '-'a > • -i- therefore it is measured by some prime number [vir. 31] Let it be measured by the prime number G. I say that G is not the same with any of the numbers A,B,C. For, if possible, let it be so. Now A, B, C measure DE ; ■" '• therefore G also will measure DE. '^\ ^, ',' ",' |' But it also measures EE. Therefore G, being a number, will measure the remainder, the unit i?^: .., . , , . , which is absurd. Therefore G is not the same with any one of the numbers A,B,C. . I , And by hypothesis it is prime. • Therefore the prime numbers A, B, C, G have been found which are more than the assigned multitude of A, B, C. Q. E. D. IX. 20-I*] PROPOSITIONS 19—2* 413 We have here the important proposition that the number of prime numiert ts infinite. The proof will be seen to be the same as that given in our algebraical text-books. Let a, ^, e, ..■ Ji be any prime numbers. Take the product aie ... k and add unity. Then {ai( ... i + i) is either a prime number or not a prime number. (i) If it /r, we have added another prime number to those given. (2) If it is not, it must be measured by some prime number [vii. 31], say/. Now/ cannot be identical with any of the prime numbers a, b,e, ... k. For, if it [s, it will divide abc ... k. Therefore, since it divides l^abc...'k.^ i) also, it will measure the difference, or unity : which is impossible. Therefore in any case we have obtained one fresh prime number. And the process can be carried on to any extent. Proposition 21. If as many even numbers as we please be added together, tke whole is even. For let as many even numbers as we please, AB, BC, CD, DE, be added together ; I say that the whole AE * b c o e is even. For, since each of the numbers AB, BC, CD, DE is even, it has a half part ,; [vii, Def. 6] so that the whole AE also has a half part. But an even number is that which is divisible into two equal parts ; [«A] therefore AE is even, Q. E. D. In this and the following propositions up to ix. 34 inclusive we have a number of theorems about odd, even, "even-times even" and "even- times odd" numbers respectively. They are all simple and require no explanation in order to enable them to be followed easily. Proposition 22, If as many odd numbers as we please be added together, and their multitude be even, the whole will be even. For let as many odd numbers as we please, AB, BC, CD, DE, even in multitude, be added together ; , I say that the whole AE is even. 414 " BOOK IX [ix. aa-H For, since each of the numbers AB, BC, CD, DE is odd, if an unit be subtracted from each, each of the remainders will be even ; [vn, ]>ef. 7] so that the sum of them will be even. > [ix. 31] * ^ p D E But the multitude of the units is also even. Therefore the whole AE is also even. [ix. ai] Q. E, D. Proposition 23. If as many odd numders as we please be added together, and their multitude be odd, the whole will also be odd. For let as many odd numbers as we please, AB, BC, CD, the multitude of which is odd, be added tc^ether ; a b c e d I say that the whole AD is ' *" " also odd. Let the unit DE be suht»^cted from CD ; therefore the remainder CE is even. But CA is also even ; therefore the whole AE is also even. And DE is an unit. Therefore AD is odd, 3. LiteniUj' '* let there ^^c as many numbers u we please, of which /tf the muLtUude it odd." Thb forai, natural in Greek, is awkward in English, Proposition 24. If from an even number an even number be subtracted, the remainder will be even. For from the even number AB let the even number BC be subtracted : 1 say that the remainder CA is even. a 9 a For, since AB is even, it has a half part, [vn. I>ef. 6] [vii 1. Def. 7] [■X. ..] t.x. "] [v.> :. Def. 71 Q. , E. D. IX. 34-a6] PROPOSITIONS aa— a6 415 For the same reason BC also has a half part ; so that the remainder [CA also has a half part, and] AC \i therefore even. • , • : . Q. E. D, Proposition 25. 1/ from an even number an odd number be subtracted, the remainder will be odd. For from the even number AB let the odd number BC be subtracted ; I say that the remainder CA is odd. /k c d b For let the unit CD be sub- tracted from BC ; therefore DB is even. • [vn. Def. 7] But AB is also even ; therefore the remainder AD is also even. [ix. 84] And CD is an unit ; therefore CA is odd. ' [vn. Def. 7] I' Q. E. D, Proposition 26. If from an odd number an odd number be subtracted, the remainder will be even. For from the odd number AB let the odd number BC be subtracted \ I say that the remainder CA is even. p^ o d a For, since AB is odd, let the unit BD be subtracted ; ... therefore the remainder AD is even. [vii, Def. 7] For the same reason CD is also even ; [vn. Def. 7] so that the remainder CA is also even, [ix. 14] Q. E. D. 4t6 o. BOOK DC :>'< [ix. 17— 39 Proposition 27. If from an odd number an even number be subtracted, the remainder will be odd. For from the odd number j45 let the even number BC be subtracted ; I say that the remainder CA is odd. Let the unit AD be subtracted ; — ^ therefore DB is even. [vn. Def, 7] But BC is also even ; therefore the remainder CD is even. [ix. ^4] Therefore CA is odd. [vil IM, 7] Q. E. D. Proposition 28, •., ..- ■ If an odd number by multiplying an even number make some number, the product will be even. For let the odd number A by multiplying the even number B make C\ „ , . I say that C is even. For, since A by multiplying B has made C, therefore C is made up of as many numbers equal to B as there are units in A. [vn. Def. 15] And B is even ; therefore C is made up of even numbers. But, if as maoy even numbers as we please be added together, the whole is even. [ix. ai] Therefore C is even. Q. E. D. Proposition 29. If an odd number by multiplying an odd number muke some number, the product will be odd. For let the odd number A by multiplying the odd number B make C ; I say that C is odd. * For, since A by multiplying B has g made C, IX. 29—31] PROPOSITIONS »7— 31 417 therefore C is made up of as many numbers equal to B as there are units in A. [vii. Def. 15] And each of the numbers A, B is odd ; therefore C is made up of odd numbers the multitude of which is odd. '''tins C is odd. ,., .r, --li-^Ki.- ' b^'ii ' Q, E. D. Proposition 30. // an odd number measure an even number^ it wiU also measure Ike half of it. • ■• For let the odd number A measure the even number B ; I say that it will also measure the half of it. ^__ For, since A measures B, a let it measure it according to C ; o I say that C is not odd. For, if possible, let it be so. ' • ^ Then, since A measures B according to C, "'- " ** therefore A by multiplying C has made B. Therefore B is made up of odd numbers the multitude of which is odd. Therefore .5 is odd : ' n ;: \ ■ v. • ["c. 33] which is absurd, for by hypothesis it is even. ■• ••- ■ . Therefore C is not odd ; < : ^ ' therefore C is even. - •'"' ^c» .our, ■ , r Thift A measures B an even number of times. For this reason then it also measures the half of it. Q. E, D. V 1 , , ,, Proposition 31. // an odd number be prime to any nutter, it will also be prime to the double of it. For let the odd number A be prime to any number B, and let C be double of B ; I say that A is prime to C. For, if they are not prime ^ to one another, some number ^ will measure them. "" 0lt BOOK IX ■ ^ ■ [ix. 31, 3a Let a number measure them, and let it be Z*. Now -^ is odd ; therefore D is also odd. And since D which is odd measures C, and C is even, "''' ''■' therefore [ZJ] will measure the half of C also, ' [ix. 30] But ^ is half of C; therefore D measures B. ■■• But it also measures A ; therefore D measures /I, B which are prime to one another: which is impossible. Therefore A cannot but be prime to C. Therefore A, C are prime to one another. Q. £. D. Proposition 52. Each of the numbers which are continually doubled beginning from a dyad is even-times even only. For let as many numbers as we please, B, C, D, have been continually doubled beginning from the dyad A ; * — I say that B, C, D are even- ' times even only. . Now that each of the numbers B, C, D is even-times even is manifest ; for it is doubled from a dyad. I say that it is also even-times even only. For let an unit be set out. Since then as many numbers as we please beginning from an unit are in continued proportion, and the number A after the unit is prime, therefore D, the greatest of the numbers A, B, C, D, will not be measured by any other number except A, B, C. [ix. 13] And each of the numbers A, B, C is even ; therefore D is even-times even only. [vn. Def. 8] Similarly we can prove that each of the numbers B, C is even-times even only. Q. E. D. VL 3a— 34] PROPOSITIONS 31—34 419 See the notes on vti. Deff. 8 to 11 for a discussion of the difficulties shown by lamblichus to be involved by the Euclidean definitions of " even- times even," "eveo-timesodd" and "odd-times even," Proposition 33. I/a number have its haif odd, it is even-times odd only. For let the number A have its half odd ; I say that A Is even-times odd only. Now that it is even-times odd is % manifest ; for the half of it, being odd, measures it an even number of times. [vil Def. 9) I say next that it is also even -times odd only. For, if A is even-times even also, it will be measured by an even number according to an even number ; [vii. Def. 8] so that the half of it will also be measured by an even number though it is odd : which is absurd. ^ , . ,, ,, Therefore A is even-times odd only, ^ , . Q. B. d. Propositioh 34- If a numier neither be one of those which are continually doubled from a dyad, nor have its half odd, it is both even- times even and even-times odd. For let the number A neither be one of those doubled from a dyad, nor have its half odd ■; I say that A is both even-times even j. and even-times odd. Now that A is even-times even is manifest ; for it has not its half odd. [vu. Def. 8] I say next that it is also even-times odd. For, if we bisect A, then bisect its half, and do this con- tinually, we shall come upon some odd number which will measure A according to an even number. For, if not, we shall come upon a dyad, and A will be among those which are doubled from a dyad ; which is contrary to the hypothesis. 4«> t.: BOOK IX [ix. 34, 35 Thus A is even-times odd. But it was also proved even-times even. Therefore A is both even-times even and even-times odd. Q. E. D. Propositiok 35. If as many numbers as we please be in coniinved proportion, and there be subtracted from the second and the last numbers equal to the first, then, as ths excess of the second is to the first, so will the excess of the last be to all those before it. Let there be as many numbers as we please in continued proportion, A, BC, D, EF, . .• , beginning from A as least, ' " a- and let there be subtracted ' ' B-^0 from BC and i^^the numbers ° BG, FH, each equal to A ; e -^ j^F I say that, as GC is to A, so \% EH \.Q A, BC, D. For let FK be made equal to BC, and FL equal to D, Then, since FK is equal to BC, and of these the part FH is equal to the part BG, therefore the remainder HK is equal to the remainder GC. And since, as EF is to /?, so is /? to BC, and BC to A, while D is equal to FL, BC to FK, and A to FH, therefore, as EF is to FL, so is LF to FK, and FK to FH. Separando, as EL is to LF, so is LK to FK, and KH to FH. [vii. II, 13] Therefore also, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents ; [vii. I a) therefore, as KH is to FH, so are EL, LK, KH to LF, FK, HF. But KH is equal to CG, FH to A, and LF, FK, HFto D, BC, A ; therefore, as CG is to yj, so is EH to D, BC, A. Therefore, as the excess of the second is to the first, so is the excess of the last to all those before it :• • Q. E. D. IX.3S. j6] PROPOSITIONS 34—36 411 This proposition is perhaps the most interesting in the arithmetical Books, since it gives a method, and a very el^;ant one. of summing any serin of terms in geometri^ai progression. Let a,, a,, a,,...a^, a,^., be a series of terms in geometrical progression. Then Euclid's proposition proves that («ii+i-Oi> : (a, + i*. + ... +a,) = (o,-fl,) : Ui- For clearness' sake we will on this occasion use the fractionul notation of algebra to represent proportions. Euclid's method then comes to this. Since ??!!=.?!_= ... = «», ' ' ^■ we have, separandi), "n*! — °ii _ ^-''n- i _ _ tfj — i»a ^ gj-tfi whence, since, as one of the antecedents is to one of the consequents, so is the sum of all the antecedents to the sum of ail the consequents, [vti. 12] <^%*\ - "1 ^ I)-"! which gives a, + «,+ ... 4- a,, or S.. If, to compare the result with that arrived at in algebraical text-books, we write the series in the form a, ar, ar',.,.ar'~' (n terms), a>* ~a ar~a we have 5, a alr'-l) s,= Proposition 36. jy as many numbers as we please beginning fr(nn an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect. For let as many tiumbers as we please, A, B, C, D, beginning from an unit be set out in double proportion, until the sum of all becomes prime, let E be equal to the sum, atid let E by multiplying Z? make FG ; I say that FG is perfect. For, however many A, B, C, D are in multitude, let so many^, HK^ L, Mh& taken in double proportion beginning from £ ; therefore, ex aeguali, as A is to D, so is E to M. [vii. 14] 4« BOOK IX [1x36 Therefore the product of E, D is equal to the product of A, M, [vii- 19] And the product of E, D is FG ; therefore the product of ^4, M is also FG. , ; r•i^'^ Therefore A by multiplying M has made FG ; therefore M measures FG according to the units in A. And ^ is a dyad ; therefore FG is double of M. .^ — A != E M F 1 Q H Q- But M, L, HK, E are continuously double of each other ; therefore E, HK, L, M, FG are continuously proportional in double proportion. Now let there be subtracted from the second /f/C and the last FG the numbers /W, FO, each equal to the first E ; therefore, as the excess of the second is to the first, so is the excess of the last to all those before it [ix. 35] Therefore, as //JC is to E, so is OG to M, L, KH, E. And NK is equal to E ; therefore OG is also equal to M, L, HK, E. But FO is also equal Xa E, . and E is equal to A, B, C, D and the unit Therefore the whole FG is equal to E, HK, L, M and A, B, C, D and the unit ; and it is measured by them. I say also that FG will not be measured by any other number except A, B, C, Z>, E, HK, L, M and the unit For, if possible, let some number P measure FG, and let P not be the same with any of the numbers A, B, C, D, E, HK, L, M, And, as many times as P measures FG, so many units let there be in i? ; therefore Q by multiplying P has made FG. IX. 36] PROPOSITION 36 413 :i But, further, £ has also by multiplying D made FG ; therefore, as ^ is to 0, so is /* to D. [vn. 19] And, since A, B, C, D are continuously proportional beginning from an unit. therefore D will not be measured by any other number except A, B, C. [«. 13] And, by hypothesis, P is not the same with any of the numbers A, B, C; therefore P will not measure D. But, as P is to Z?, so is ^ to i2 ; therefore neither does £ measure Q. [vii, Def. ao} And £ is prime ; and any prime number is prime to any number which it does not measure. [vn. >9] Therefore E, Q are prime to one another. But primes are also least, [vii. *i] and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent ; [vn, 30] and, as £■ is to 0, so is /* to Z? ; therefore £ measures P the same number of times that Q measures D. But D is not measured by any other number except A, B. C; therefore Q is the same with one of the numbers A, B, C. Let it be the same with B. And, however many B, C, D are in multitude, let so many £^ HK, L be taken beginning from £. Now £, HK, L are in the same ratio with B, C, D\ therefore, ex aequali, as ^ is to D, so is ^ to Z. [vn, 14} Therefore the product of B, L is equal to the product of A E. [vii. 19] But the product of D, E is equal to the product of Q, P; therefore the product of Q, P is also equal to the product of B,L. Therefore, as is to B^ so is L to P. [vn. 19] And Q is the same with B ; therefore L is also the same with P : 434 BOOK IX [ix. 36 which is impossible, for by hypothesis P is not the same with any of the numbers set out. Therefore no number will measure FG except A^ B, C, D, E, HK, L, J/ and the unit. And FG was proved equal to A, B, C, D, E, HK, L, M and the unit ; and a perfect number is that which is equal to its own parts ; [vii. D«f. 2»] therefore FG is perfect, Q, E. i>. If the sum of any number of terms of the series I, i, a*, ... 2""' be prime, and the said sum ( e multiplied by the last term, the product will be a "perfect" number, i.e. equul to the sum of all its factors. Ijet I + I + j' + . . . + a""' (= 5,) be prime ; then shall S^ . 2""' be "perfect," Take {it - i) terms of the series •->ii> ^"Jii, 2 Ob, ... i Ob. These are then terr.is proportional to the terms a, 3', 2', ... 2"-'. Therefore, ex aequali, . , jra— = ^;a-^j;, , [vii. 14] or 2 . 2"-'5',= j"-' . J,. ■ [vii. 19] (This is of COttrse obvious algebraically, but Euclid's notation requires him to prove it.) Now, by IX. 35, we can sum the series S^ + 2 j^ + ., . + 2"-'i^, and (25. - S^) : 5. = (a-' 5. - .S;) : (.S; + a^; + ... + a—3.). Therefore .S; + 25. + 2»5, + . . . + 2*-'5, = a"-'5, - S,, or s"-"^, = 5, + a5. + 2*5, + . . . + 2*-*Sn + J. = S^+tS„+ ... + 2'-»5, + (i + 2 + 2' + ... 4- a"-"), and 2""' 5, is measured by every term of the right hand expression. It is now necessary to prove that a*"'.?, cannot have any factor except those terms. Suppose, if possible, that it has a factor x difTerent from all of them, and let 2'"''S'» = x . ut. Therefore S^^: m = x : 2""'. [vii. 19] Now 2"~' can only be measured by the preceding terms of the series I, 2, a',.., a"-', [IX. 13J and X is different from all of these ; j _ therefore x does not measure a""', so that S, dots not measure m. , [vii, Def. 20] And S, is prime: therefore it is prime to m. [vii. 29] It follows [vii. 20, 2r] that m measures a""'. IX. 36] PROPOSITION 36 4^5 Suppose that > <■ w = J^ .i- Now, ex aequaii, a' r i""' = S, : j"-'-' 5,. ' ' Therefore a'' . a"""-' 5« = s""' 5. [vji. 1 9] - X . m, from above. And m = a'' ; therefore .r = 2"-'-' S», one of the terms of the series Sx,^S^,2*Sf, .,.i*~'S^: which contradicts the h)pothesis. There a"~'5, has no factors except 5„ zSn, a'5„ ... i"-'^., I, 2, a', ... i'-\ Theon of Smyrna and Nicomachus both define a " perfect " number and give the law of its formation, Nicomachus gives four perfect numbers and no more, namely 6, 28, 496, 8138. He says they are formed in "ordered" fashion, there being one among the units {i.e. less than 10), one among the tens (less than too), one among the hundreds (less than 1000) and one among the thousands {less than loooo) ; he adds that they terminate in 6 or 8 alternately. They do all terminate in 6 or 8, as can easily be proved by means of the formula (2"- 1)1""' {cf. Loria, Le tcienst tmtte neW antica Grtda, pp. 840 — i), but not alternately, for the fifth and sixth perfect numbers both end in 6, and the seventh and eighth both end in 8. lamblichus adds a tentative suggestion that perhaps there may he, in like manner, one perfect number among the "first myriads" (less than loooo'), one among the "second myriads" (less than loooo'), and so on. This is, as we shall see, incorrect. It is natural that the subject of perfect numbers should, ever since Enclid's time, have had a fascination for mathematicians. Fermat (160 1—1655), in a letter to Mersenne {CEuvres de Fermai, ed. Tannery and Henry, Vol. 11., 1894, pp. 197 — 9), enunciated three propositions which much facilitate the investigation whether a given number of the form a"-! is prime or not. If we write in one line the exponents i, a, 3, 4, etc. of the successive powers of 2 and underneath them respectively the numbers representing the correspond- ing powers of a diminished by i, thus, 1334567 S 9 10 II ...n I 3 7 IS 3' 63 "7 ass S" >«23 2047...2"-i, the following relations are found to subsist between the numbers in the first line and those directly below them in the second line. I. If the exponent is not a prime number, the corresponding number is not a prime number either (since a" ~ i is always divisible by a' — i as well as by a' - 1 ). a. If the exponent is a prime number, the corresponding number dimi- nished by I is divisible by twice the exponent, [(a*- a)/2M = (a""' -i)/« ; so that this is a special case of " Fermat's theorem that, if/ is a prime number and a is prime to/, then cf~' is divisible by/.] 3. If the exponent n is a prime number, the corresponding number is only divisible by numbers of the form {tmn+ i). If therefore the corre- sponding number in the second line has no factors of this form, it has no inte>;ral factor. The first and third of these propositions are those which are specially useful for the purpose in question. As usual, Fermat does not give his proofs but merely adds : " Voilil trois fort belles propositions que j'ay trouvees et prouv^es non sans peine. Je les puis apfteller les fondements de I'invention des nombres parfaits." 496 ^ BOOK IX [ix. 36 I append a few details o( discoveries of further perfect numbers after the first four. The next are as follows : fifth, 3"(3''-i) = 33 SSO 336 •' f sixth, »" (*■'-!) = 8 589 869 oj6 seventh, i"(i'*-i)= 137 438 691 jaS eighth, 2* (2"-!)= 2 305 843 008 139 95* ia8 ninth, 2" (2*' - r ) = 2 658 455 99 1 569 83 1 744 654 691 6 1 5 95 3 842 1 76 tenth, a"(a"-i). It has further been proved that s'"- 1 is prime, and so is a'"- r. Hence 3'" (a'"-!) and 3'"(2"'-r) are two more perfect numbers. The fifth perfect number may have been known to kmblichus, though he does not give it ; it was however known, with all its factors, in the fifteenth century, as appears from a tract written in German which was discovered by Curtze (Cod. lat. Monac. 14908). The first eight perfect numbers were calculated by Jean Frestet(d. 1670). Fermat had stated, and Euler proved, that a"- I is prime. The ninth perfect number was found by P. Seelhoff {Zatithrift filr Math, u, Physik, xxxi., 1886, pp. 174 — 8) and verified by E, Lucas (Mat/Usis, vii,, 1887, pp. 45 — 6). The tenth was discovered by R. E. Powers (see Bulletin of ifu Ameriam Mathematical Society, xvni,, 191*, p. i6a), 2"*— I was proved to be prime by E. Fauqueuibergue and R. E. Powers (191 4), while Fauquembergue proved that 2""-! is prime. There have been attempts, so far unsuccessful, to solve the question whether there exist other " perfect numbers " than those of Euclid, and, in particular, p>erfect numbers which are odd. (Cf. several notes by Sylvester in Comptes rtndus, cvi., t888 ; Catalan, " Mdlanges mathrfmatiques " In M^m. de la Sec. dt Liige, 2* S^rie, xv,, 1888, pp. ao5 — 7 ; C. Servais in Mathisii, vii., pp. 228 — 30 and VIII., pp. 92 — 93, 135; E, Cesiro in Math^sis, vii., pp. 245 — 6 ; E. Lucas in Mathisis, X., pp. 74 — 6). For the detailed history of the whole subject see L. E. Dickson, History o/the Theory of Numbers, Vol. 1., 19 19, pp. iii — iv, 3 — 33. INDEX OF GREEK WORDS AND FORMS. dxpot, extreme {of numbers in a series) 3^3| 367: Axpor xol fUvof \byv rtTfi^ffOu, *'to be cut in citrcme Ewd mean ratio*' 1S9 aAo-yoj, irrational 117-8 iraXoyiSj proportiiM); definition* oft inter- polated 1^9 dviXjryovsdfd \&yov, propoTtLonal or in pro- portion: used as Lad^clmable adj^ and as adv. r^pt 165: pjiffn drdkoyof^ mean pro- portional (of straight line) 119, similarly ^ff« AwA^aryor of numbers 195^ ^6^ etc. : rpfrq (rpJrot) 6*6Xey», third proportional 4I41 4O7-S: TtrAfT^ [riritfiTtH) AtdXoyoi/j fourth proportional 115,409: ^f 1^ drdAo-^op in continued propottion 346 4j'd'rd?\^»{}<,6yot)j inverse (ratio) ^ inversely t34 dya^Tpwftij XA7W, conversion of a ratio 135 dFi^dxit drurdjTLt Cirdt, unequal by unequal by equal (of solid numbers) = scaltrtUt t^fjwlvKojf ff^xiffKot or pvtd^Ktn 190 dra^iifr TtrayfUrvr TUtr ^^yvf (of perturbed prGp^tiQn) in Archimedes 136 d»Ta>aJ!pca'LTt i4 <l^7H>|i> definition of SQftti raiia in Aristotle (d^tfu^o^^ii Alexander) no: tcrmi expLained m dmirEirorf^^TtL v^^^i^juara, reciprocal (^reci* procaily related) figures, interpolated def. off 189 dirXamSt, brcadthlcss (of prime numbers) 185 dTOJcarHTartjcAt} neumni {=jpheri£al}, of numbers 391 drrftftfoff to rrneti occasionally to tottiA (instead of i^vrtffffm) ^i a]so=to/arf ikrpHgh, to ii> tfA 70 dpiff^, number^ definitions of^ 380 i^i&Kii Apviot^^.flOl^ (Nicouiachus) 183 dpr^dnr i/)T4<Hj £vtn-timts even i8i-a d/mdivtr irf/R(ro-d(^ even-times odii ^83-4 4/moF^«TTot> tfFm-tfdtf(Niconiacbusetc.) iS* 4>)Ti(>i (d^^^i)^ ev«n (number) 181 4«^£p0rmj (prime and) locomposite (of numbers) 184 ^f^jr^nt, to stand {of anfle standing^ on circumrercnce} 4 ^fiiffxor, ^itiir-skaped (of "scalene" solid niunben} 990 7ryor^B(in constnictionB), **l«t it be made" ^48 Tfywif Jbf rfi) Ti iirtTaxW"! "what was en- joined will have been dune" Bo, itit 7c»A^»Dtt 4 ^£ n^ur, *' their proditet** 316, 336 etc,: h Ik toD ^pit y(?fi;iej'Of='* the tptare of the one " 317 ywi&fivrt gnomon: Democritus irf/4 Atatpo- p^ 7ifc3Jiai'« (T^tiitiiji or yufliit}) ij npt ^atStf-jDt jti^jcXov Kol f^oipift 40: (of numbers) 189 -/pafifitK6f, linear (of numb^ in one diinMn-> ston) 1S7: (of prime numbers) 185 ypdiptv ffatt ^'lo be proved" (Aristotle) tta St&rtpot, see&nd&ry (of numbers): in Nico- machus and lamblichus a subdivision of odd 186, 387 j^fl^^^ycw, **admittLnf ^' (of segment of circle admitting or contaming an angle) 5 Jnu/Kr^rtfoj (used of ^^ separation" of ratios): jfoipf^^n-a, seporand^y opp* to t\^KttfiX¥v,t £&mpcn£nd& 16B itatptffit \^you, separfttiott, literally division^ of ratio 135 ^inimrffUrTi {ifia\oyla), disjoinedi = discrete (proportion) 2^^ ii^HwTi, separanda^ literally dividends {cH proportions) 135 ii^py^iUt^ (draXfi^^a), discrete (proportion)^ i.e, in four termj, as distinct from continuous {avwt.x¥i <nmj/i^Fif) in MrM terms [3i;| 393 ii-ffxB^ (AidytfLp), "let it be drawn through** or *' aerass** 7 &^ {!(r0Uf«: d/i^va/t (of ratios) 156: i^i' trov ^ rrrapayfi^nff draAc^^^, ^Vj<^ aeffuati in per- lurbed proportion" 136 JtffiXoupof, twice-iruncaied (of pyramidal numbers) 391 A^TAdfTioT \6y<ftt double ratio : ffiK-Xatfibv X^^ott dupiicaie ratio, contrasted with, 133 Sf^a^r^ posver: =actual value of a aub- multiple in units (Nicomachus) i8t ; ^side of number not a complete square (i.e. roctt or surd) in Plato ^88^ ^90: =square in Plato 194-5 fWw, figure 134: =form 254 fvotfroff each: curious use of^ 79 fWiiptfiA, defect (in application of areas) 161 iXkeiww^ *"fall shtirt" (in application of area$) i((i 4^ INDEX OF GREEK WORDS AND FORMS tftMi-rr^tM^ /all in ( = bc interpolated) 35B ba rXe^, ''several fftut" (def, of tiiunber) /roXMf (\4yM),altenute {ratio): aitemntely* &iternand& [14 ^op/i^if^ ^^pi >'» (active) I IV, Def. 7 «nd IV. I, 7<h 80, &i jrrb, within (of internal contact of cinles) 13 . . ^^ ^dXo7or} in continued proportion (of tcnn^ in geometrical progression) 546 ^ri^jHor X^TQii suptfpartuularU ration = the ratio Jb+ [): Bv 195 ^irfiTffAot (4^0^}, plane (number) 387-S ^6^ra, conse^^uenla ( = "foUowing" terms) in a proportion 134, ijS ire/Mjiii)jriff| obiong (of numbers): in Plato = r^>4^T^, whicb however is distinguished from /rt^^^Kijf by Nicomachus etc. 189- t06vypafituii&tf rectilinear (term for prime numbers) ^85 *d*y/itT^ifdf, eulhymetric (of primes) ^85 ^^fX^^faT antecedents ("leading" terms) in a proportion (34 TWtpt than : construction after 6tr\aait«f etc. tSw/iiiir^t of jsquare number (lamblichus) 193 iff^tt tffdxit Iff Off equal multiplied by equal and again by equal (of a cube number) 490, ^gt Itrim tttojt equal multiplied by equal (of a square number) 391 fo-cUif tffoj 4kKTT0*ij(tt (^ifbpint), species of solid numberst = T\ip0lt [Svitt or mfMT) KoKttffBvj "let it be calledt" indicating origi- nality of a definiUon 119 KaTOfLerptuf^taeAttite uj: without remainder, completely {irXifpovrTaH] 380 itaTauKtvd^Ut construct: rCtr aOrvf ttaTa- 7«CMi(r#^Twv^ *' with the ^me construc- tion *' 1 1 KararotA^ irap^^t, Secfh cantmis of Euclid ttirrpowi centre: tj Ik rw K. = radiui 2 KtptxTOtiSTi^ ywpiokt hortttiki angle 4, 39, 40 JcXaCt to *rff*t^j iij^fiT/! ffCKXi^rfffj ffi^ irdXcf 47 ; HjrXdd-tfcu^ def- of^ alluded to by AriF^olIf: 47 xMoupof, trufKoied (of pyramidal number mitmi vertex) igi Kvg\u6t, cyciici a particular species of square number 191 Xd^ot^ ratio: meaning 117 : definition of, 11&-9: original meaning (of someLbing €xpressed) accounts for use of AXoT^ri having no ratio, irraiiatial 117 fu^ov^6a.i^ to be isoIiUidf of ftori^i (Theon of Smyrna) ^79 fiipofj part; two meanings 115: gencrallys submultipte 380: m'^, parfj (^proper fr«ction) 115, 180 fii^ AriXtyot (td^cla), fJrof driXoyof {Api9* fiM), mean proportional (straight line or number) 119, 195, 363 etc. fi.ii ydffy "suppose it i> not" 7 fi^Kot, length (of ntimber in one dimension) : = side of complete square in Plato 188 tu»Att unit, moniad: supposed etymological connexion with fi6m, solitary] ftot^t rest 379 5^i«; similar: (of lectilinod figures) 188: (of plane and solid numbers) ^93 ^fiiotAnfs Xiytaft '^flimilarit^ of ratios" (mier- polated def. of proportion) 119 ifiiikayoi^ homologous, corresponding 134; exceptionally "in the same ratio with" Spat, Urm, m a proportion 131 rapo^dXXet^ ^70, used, exceptionally, instead of wapa^AWfiy rapk or draypd^civ A.t6 161 irupaXXdirru, *'faJl sideways" or '*awry" 54 mvTAypafiitAoy 99 wtpalr9wa twAttp, "limiting quantity" (Thymaridas* definilion of unit) 179 vepi^t/d^Kij ApriQi, edd'iirttii tvcn 181-4 ^tpuj^ibcit irtpiffff6i, odd-timts odd 184 r«^(r4pT-mi^ otkleven (Nicomachus etc-) 3B3 ■rtputv^, odd tSi irifMjraft how grei^t: refers to centifttdom (geoTneErical) magnitude as woff&i loditftvte (multitude) 116-7 T^SiKirift, used in v. Def. 3, and spurious De^^ 5 of V] . : = n^ (not quanhipiiiiiy as it is trarulated by Pe Morgan) 116-7^ 1S9- 00: supposed mukip1icationofi-)t\ur^iTi7Tct \vu Def. 5) r3i; dislinclion between T7Auifin7f and fHytB^i 117 rXdrfrt, breadth: (of numbers) 1S8 irXeirpd, side: (of ^tors of "plane'' and '* solid" numbers) 188 vXi^ti^or uptfffiiiitov or rtr^paa^ixdur^ defined or finite multitude (delinition of number) tSo: 4k ^tofdlup ffvytelfunv irK^Btn (Euclid's def.) 38a r^XXarXuridj'f^r, multiply : defined tBj xoXXavXaffiOff^df I multiplication : itad' 6wotof- Qijr v4XXarXwr<a^i4^ "(arising) from any mukiplJcjLtion whatever*' no iroXXairXd<riaft multiple ; i^Ajcit roXXurXdirca, equimultiples 110 etc. Trok6r\tupor, multilAtcrah excludes Ttrpir rXfupoi^, quadrilateral 339 iropitrciffdatt \o ^ttd 148 irod'tijciT tfurAKii -roo^i "so many times so many limes so many*' (of solid nambers, in Aristotle) i86t 1190 •r9oi.Ki.% TTOffot^ ^'so many times so rruny^' (of pbne numbers, in AHstotle) 186 TiKT^i qvantily, in Aristotle iij: refers to nmltimde as t^Mkoi' to magnitude 1 16-7 INDEX OF GREEK WORDS AND FORMS 429 TTpQfiiitnti, obhnj; (of numbers): in Plato = ^*^^ifijj» but disiinguished by Nico- iTuchus etc. ^89-901 iQ.i TpHra'a7^^«, tOdiiraicrm At: (of a circle) to ismpUti^ when segment is given 56 vpcvfifptii'i to £nd in addition (of finding tbird and fourth proportionals} 114 r/Ktffor, prime 1*4-5 TpuTDi r^ dWi^Xot^T, (numbers) prime to one anotber 4B5-6 ^ifr^, rational (literally '* expressible") T17 "J'rcnir continuous : ^vj^x^ d*ttXoyft| '* oon- tinaoui proportion" (in three terms) 131 runf^jL^i^ duHiXryia, ccnmcted {i-e. con- tlnuotis) proportion 131^ «j : <fwijft^rof, of iompsmHa ratio in Archimedes 1^3 ffVitBirrt, compOMnda 1 34-5 rirr0f<rit XE^yair, composition of a ratio, dis- tinct from iampoundtHg ot ratios 134-5 fft^fffTflT, composite (of numbers) : in Nico- machus and lambUchus a subdivision of odd 366 tfittrtfif/Uj n^xei^oi {of ratios) 135, iSg-90: ffvyifi^^PR and ^taip^0iw7Vk {c^mpontndo and ji^mH^) u^ed relatively to i^ne another fftfffTij;ia fi/^SuTr "collection of units" (def. of number) a So ffvimffiaTtK6tf collective 379 ff^<u^K6u spMerjca/ (of A particular spiecies of cube dumber) 391 f^it^LvKot, or ff^rjtrlffMoi, of solid number with all three sides unequal (= '* scalene") 7tp ffX^ffu* ** relation '* ; wwA rxi^rni '* a sort of relation" {in def* of ratio) ji5-7 TaA-np4icifl^ of square number (Nicom,) ^95 TaCr&nji \&yuWi "sameness of ratios" 119 riXfioti pfr/ec/ (of a class of numbers) 193-4 TnTat^ivyj {iiiahoyla), *' ordered (proportion) Ttrttpayttinf AraXoyiAf perturbed proportion 136 TfrpdrXtu/Kii', qu4dril*teral» not a " polygon" Tfi^jCia (Ai>H\«;)t segment {of circle): r/i^jmirn 7C4*'fa, angle of ^ segment 4; I9 r^^juan 7urr^ angle in a segment 4 Tp^e^ (ffi[!jf\«'}f sector (of circle): (TKurvrft- ^wftr Tfl^^, "shoemaker's knife" 3 r^^At^l-^i i^i figure}^ mtor-likt 5 rG<ravrarXdiriov, "the same multiple** 146 TpLyijror : r^ rptrKw*, rh ^* dXX^Xur, ttiplei inttrwoven triangle, = pentagram 99 TpLv\a.trt^^ triple, TpiTXnffW, triplicate {of ratios) 133 riffX^^^^t happen: 4XXa, cC frio(<c» £(ruia To^Xaa-Xd^fai "other, chance, equimulti- ples" 143-4^ ryxoLtfii 7i*if(«» "(jwi'ttigle'* uirr^<Xif$ or virrprAtuff, "over-perfect" (of a class of numbers} 193-4 (iro3(TX«rtor, st^dupiicate^ = half (Nico* machus) «8o AraT«X\iiYX^u)r, submulttple (Nioomachus) iSo 0^r» height [89 Xu^fVt av^a. 354 ENGLISH INDEX. Adnutai 199 Alcinotu 96 AiUms/e and a/temateiy (of ratiiK) 134 AUeituttire prtxifa, interpoUtcd [a. uu 9 utd following) 11 : thmt in ui. to cUiuked by tttron 93-4 Amaldii LJgt>, 39* 11$ Amdipi^tii eate of VI. y, joS-y ADuiimander 1 1 1 Anaximenu iii Angle : in|r!es not leu thin two right an^^les not recogni$«d as tingles (cf. Heron, ProcluSf Zenodorus) 47-9: haUcw^angied figure (the le-cnLiant uigle was exf^n^ 48; dia Eudid eictend ^^ angle" to angles gremter than two right angles in VI. 33? *7J-6: '* angle a/ semicircle" and " t^f legment" 4: herniiif ttig\t 4, ^9, 40 : controversies a.bont *' an^le of semicircle ** and ^bm/i^f angle 39-41 (see alsto JfernUki) AtUue^nti (le^ing terms in proportion) i J4 AiidparalLels : may be used for construction of VI, II, 115 Apolloaius : ^/iaixf firvmri problem fromi&T, lemma by Pappus on, O4-5 : FUnt Loci, theorem from (arising out of EacL vt, 5), also found in Aristotle 198-900: 75, 190, '*? . Applicatii>n of areas (including txcitdmg and jaiimg short) corresponding to soLutioo of quadratic equations 167, 358-6o» ^63-5, 166-7 AppToaimations t 7/5asappTOKimattDn Xo^J^ (Pythagoreans and Plato) 119: appruxi- mations to J^ in Archimedes and {in sexagesimal fractions) in Ptolemy 119: to T (Archimedes) j 19 : to V4300 (Theon of AleEandria) 119 Archimedes: MetAod <j{, 40: Li&tr aifump- ieruiHi proposition from, 6^ : approxima- tions to 1^3, square roois of large numbers, and to V 1 19: extension of a proportion in aymmensurablesto cover inoommcnsur^bles Archytas : proof that there is no numerical geometric mean between n aiid m Hh i 495 AnMotle; indicates proof (pre- Euclidean) that angle in semicircle is ri^bt 63 : on def. of same ratia {=sami drrara^wtf) I90*i : <m proportion as ' ' equality of ratios " 1 19 : on theorem in pfoportion not proved pfMfra//y till bis time iij: on proportion in three terms (irurfx^^i continuous)^ and in four terms (0ivpfM^« discrete) I3r»l93 : OD aitimeUt ratios 134: on jflwr^c ratio t34i 149- on similar rectilineal figures 18S: has locus- theorem (arising out of Euct- vi. 5) also given in Apollonios' Phn^ Loci 198-900: on unit ^70: on number 180: on non-applJcability of arithmetical proofs^ to magnitudes if these are not numbers 1 1 j : on deAnitions of odd and even by one another i%i'. on prime numbers 384-5: on composite numbers as plane and solid 986, 18B, 990: on representation of numbers Ly pebbles forming figures 186 Arithmetic, Ettnunts of, anterior to Euclid 195 Auguit, E. F, 33, 3£r 149, 35S, 456, 419 Austin, W. 174, tSSt in, 959 AxioBU tacitly assumed * in Book v« 137 ^ in Book vii. 994 Babylonians iii .,_ ^ , Baermann^ G, F« 913 Baltur, R. 30 Barrow: on EucL v. Def^ 3, 117: on v* Def. 5. "i: i^t i36, 938 Billin^tey, H< 56, 138 Boetbms 195 Borelli, G. A. 9, 84 Breadth (of numbers) = second dimension or factor 48^ Briggs, H. 14^ Camerer, J. G, 99, 95, 9S, J3» 34, 40, 67, jui, ijJ» 189, 913, «44 Campanus 98, 41^56, 9a, n^t '19. isi^ 1461 )8q, an, 934, 933,353^ *7S, 5w, l^if 318 CandaLla 189 Cantor, Monti 5*. 40, 97 Cardano, Hieronimo 4I Case\ Greeks did not in/tr limitii^r caseSf but proved them separately 75 Casev, J. 197 Catalan 496 Cesiro, E. 416 " Ckanct equimultiples^' in phrase ** other, chance, et^uimattiples *' T43-4 Circle: deJtnition of equal ctrclcs 9: circLeii 43* ENGLISH INDEX tmichiH^, meaning of definition, 3 : *^ circle*' in s^nse of " drcnrnfcrence ** ij; circles tntcTsecMng and tonching, difficoltiej in Euclid's treaiment of^ ^5-71 18-9, modem treatment of, 30-^ aavius *, ^i, 41, 47, 49. 53, ^^6, 67, 70, 75, 130. i79t tgo, 331, 138, 344, 171 Commandinus 47, 130^ 190 Ccmp^n^tidQ [avrffh^iy^ denoting '^composi- tion " of Ffttios f*v. '. ccmp^tunda and stparando nsed relatively to each other 168, 170 Composite nnmbei^t in Kuclid i36: with EucL and Theon of Smyrna may be even, but with Nicom. and Iambic are a sub- division of odd t86: plane and solid numbers species of* lS6 ** Composite to one another*'^ (of numbers) 586~7 Composition of ratios (ffiSvflrtftt XAyoir), de- noted by ttunpimfndo [fftn^^i^ri), distinct from compounding ratios 134-5 Compound ratio: explanation of, 133-3: questionable definition of^ T&9-io: com- pounded ratios in v. 90-'>3, 17&-8 Constqitmti {"following" termi in propor- tion) 134, 138 Ccntinucus proportion [ffv*rxift or ^uvi^^f^v^ diuXpyia) in three terms 131 Conversion of ratio (d»urrfH»^7r Xd^ou), de- noted by f^nvert^ndo f^uuTTpi^ffatTi) 135 : ionvertittdo theoreni not eat^blished by v. 19, Por. 174-51 hut proved by Simson's Prop. E 175 C&twcrtgndi> denoting " conversion " of rati<}5, €orr€ipcnding magnitudes 134 Cube: duplication o^ reduced bj Hippo- crates to problem of two mean pro- portionals 133 : cube number, def, of, 191 1 : I wo mean proportionals between two cube numbers, 194^ 364-f Curtze, M. 446 CysUii ofa particular kind of aqtiare number CymtmUhia of teotaud 44 Deda of Euclid \ Def. 1, 348 : Prop. 8, 349- 50: Prop. 44, 346-7: Prop. 55, 354: Props, 50 and 68, 349: Prop, f8| 363, 36 j : Props. 59 and 84, «C6-7 : Prop. 67 assumes part of converse of Simson^s Prop. B (Book vl) 3«4 : Prop. 70^ 750: Prop* 85, 364 : Prop. 87, 338: Pirop, 93, %%i Dechales 359 Dedekind's theory of irrational numbers ctirresponds exactly to Euch v. Def, 5, 134-6 Democ ritu & : " On diferermt e/gnfff/ien * * etc. (? on *^ angle of contact ") 40 : on parallel and infinitely near sections of cone 40: slatedj without proving, propositions about volumes of cone and pyramid 40 De Morgan, A. : on definition of ratio 1 16-7 : on extension of meaning of r-titii> to cover incommensurables 118: means of expres<i- ing ratios between incommensurables by approximation to any extent 118-9: de- fence and explanation of V. DeC 5* '^^-4: on necessity of proof that teste for greater and less, or greater and equal, ratios can- not coexist 130-1, 157 : on compound ratio [33-3, 334: sketch of proof of existence of fourth proportional (assumed in v. 18) 17 1 x proposed (emma about duplicate ratios a^ alternative means of proving vi» 33, 146-7 l 5( 7, 9-10, II, 13, 30, 33j 39, s6, 76-71 83, 101, 104, U6-9, 130, 130* i39» *+5» 197, 103, 117-81 '3«* '33* '34p »?'*■ *75 Dei'cyllides li i Dickson, L^ E. 436 Dhrismus for solution of ft qumOratic 159 DistnU proportion, di^pTifU»it ot Sitj^tvyft^rif draX^^Ja, in four terms 131, 393 *^ Dissimilarly ^rdir%d ^' proportion idro^lur rrr^'^pAwiay rmv Aiywf) in Archimedcii =r * * perturbed proport ion " 1 36 Dividittdfi (of ratios}, ae Separation, upar- ando *' Division (of ratios)/^ jw Separation Divisumi {qfjSpurij]j On, treatise by Euclid, proposition from, 5 Dodecahedron : decomposition of faces into elementary triangles, 98 Dodgson, C L. 48, 375 Duplicate ntio 133: iliirXturJtfv, duplicate^ distinct from ttrXwws, double ( = ratio 1:1), though use of terms not uni Form [33: "duplicate^" of given ratio found by VJ- 11, 3E4: lemma on duplic;tte ratio as aU tertiative to method of vi. 33 (De Morgan and others) 143-7 Duplication of cube : reduction of, by Hippo- crates, to problem of finding two mean pToptortionals [33 : wrongly supposed to be alluded to in Timaeus 33 A, B, 394-5 n. Egyptians 1 1 3 : Egyptian view of numher 180 Ennques (FO and Amftldi (U,) 30, 116 Equimultiples: '*any equimultiples what- ever/' iffcUtr raXXttTXafft* Kotf hiri>iQiK>w Trdk\aT\a<na(Tfiioy no: stereotyped phrafie "other, chancet equimultiples" 143-4: should include eiKt each magnitude I45 Eratosthenes : measurement of obliquity of ecliptic (13° 51' 3o")- 111 Escribed circles of triangle 65, 86-7 Kudemus 99, in Eudoxus 99, 180, 195: discovered general theory of proportionals covering incom- mensurables [13-3 : was first to jirove !»cicntificaUy the propositions about volumes of cone and pyramid 40 Euler 436 Eutocius: on '* vi. Def. 5" and meaning of wij\iK&rqr iiflt 13I1 189-90; gives locus- theorem from Apollonius' Plsru Loci 198- ^00 Even (number) X definitions by Pythagoreans and in Nicomachus ^St : definitions of odd ENGLISH INDEX *$i and «ven by one another unscientific {Aristotle) i8t : Nicoin. divide* eien into three e1aB$es{l)£2tfH*/fM/j«ivn and (i)^*^- tinii't a/d as extremes, and (j) tid-iima evtn as intemiediflte ^St-3 Even-times even* Euclid's use differs fit>m use \ij Nicomachun, Theon of Smyrna and lamblicbus j8i-a Evtn-times odd in Euclid different from even- wA/of Nicdmacbus and the lest iSi-^ Ex aeptalit of ratios^ t j6 : ex ae^imU pro* positions (v. 30, 11), and ex ttegvaU ^^in perturbed proportion" (v. 11, 53) t76-8 Faifofer 116 Ftuquemberguei E. 41^ Fermat, 4Q5, 416 Fourth proportional : assumption of enistence of, in V. iB, and altenuitive methods for avoiding {Saocberi, De Morgan, Simsoti. Smith and Bryant) 170-4: Clavius made the assumption an aiiotn ijo: sketch of proof of assumption by De Marfan fji : condition for existence of number which is fourth proportional to three numbers 409-1 1 Galileo Galilei; 00 iiiigit «/ emtaet 41 Geometric means ^^jsqq*; one mean between sqiuire numben 194, 36J, or between wmilar plane numbers 3;i-i : two means between cube numbers 194, 3^4-5, or between similar solid r umbers ^73-A Geometrical progression 346 sqq.: summation ofn terms of (IX. 35) 410-I GheranJ of Cremona 47 GntHnon (of numbers) 389 Golden section (flection in extreme and mean ratio), discovered by Pytbagoreani og: theory carried further by Plato and £u- doxua 99 Greater ratio: Euclid's criterion not the only one 130; arguments from greater to le*s ratios etc. unsafe unless they go back to original delinitioas (Simson on v. lo) i tfi-1 1 test for, cannot coexist with test for equal or less ratio 130-1 Greatest common measare : Euclid's method of finding corresponds exactly to ours 1181 ^99: Nicomachusgivesthesamemethad3oa Gregory, D. 116, I43 Hiibler, Th. loiM. Hankel, H. 116, ti; Hauber, C. F. 144 Heiberp, J. L. fiittim Henrici and Treutlein 30 Heron of Alexaivdria: Eucl. It). 11 interpo- lated from, 38: extetidslll.io, It toangles in segments less than semicircles 47-8: does not Tecogr\i5e angles equal to or «eater than two rightangles 47-8; proof oflforroula for uea of triangle, ti ='Ji (i - o) (r - i) {1 - 1) Bj-8: s. l6-t7. »4t >8, 34, 36, 44, iifi, 189, 301, 310, 383. 39S liippasus 97 Hippocrates of Chios i^^ }iorniike angle (vriMTtKtJ^ yvrla) 4, 39, 40: ^ffm/jilf angle and angle ^ semicircle, con- troversies on, 39-41: Proclus on, 39-40: DemtKritus may have written on hornlike angle 40: Campanus ("not angles in same icnse") 41: Cardano (^win/iVi'fj of different orders or kinds): Peletier (hertilikt angle no angle, no qoanltty, nothing ; angles of d^ semicircles right angles and equal) 41 : Clawius 41: Vieta and Galileo ("angle of contact no angle ") 41 : Wallis (angle of contact not incHnaiion at all but degree eif curBoiure) 43 Hultsch, F. 133, 190 lamblicbus 97, 1x6, 179, aSq, s8i, 3ti3, 384, iSj, 186, 187, iSS, 189, 19a, 391. 393, 193, 4i9> 41.S1 4»* Icosahedron 98 Incommensu rabies: method of testing incom- mensurability (process of finding G.C.M.) 118: means of expression consist in power of approximation without limit (De Morgan) tig: a]^roximattonK to tji (by means of side- and iAg^tntii/-num^n) 119, to ^3 and to T, 119: to V4300 by means of sexagesimal fractions 1 19 Incomposite (of number) = prime 184 Ingram!, G. 30, 116 Inverse (ratio), inversely (drdnAtr) 134: in- version is subject of v. 4, For. (Theon) 144, and of V- 7, Por. 149, but is not ETly put in either place 149: Simson's B on, directly deducihle from v. S, '44 Isosceles triangle of )V. 10: construction 01, by Pythagoreans 97-9 Jacobi, C. F, A. 188 Lachlan, R. 11G, 137, m-6, 147, !£(!, 171 Lardner, D. 38, 35 j, 171 Least common multiple 336-41 Legend re 30 : proves VI. t arid similar pro- positions in two parts (1) for comment su rabies, (1) for incommensurables 193-4 Lemma assumed in vt, 1%, 141-3: alternative propositions on duplicate ratios and ratios of which they are duplicate (De Morgan and others) 143-7 Length, ^%« (of numbers in one dimension) 587: Plato restricts term to side of inte- gral square number 167 Leotaud, Vincent 43 Linear (of numbers) =( I ) in one diraenston 187, (i) prime 185 Logical inferences, not made by Euclid aa, 19 Loria, G. 435 Lucas, £. 436 Lucian 99 Means : three kinds, arithmetic, geometric and harmonic 193-3: geometric mean is 434 ENGLISH INDEX '* pio^tAoD par exieUetue'' (ki^^im) 191-3: one geojuetnc mem between two square numbcrSj two between two cube pumbers (Plato) 194, 363-5: one geometric mean between simiUr plane numbers^ two be- tween similRT solid numbers 57'~5- po Dumerical geometric mean between n and ff + 1 {Archytas and Euclid) 195 ModerfttuSi a Pytbaeorcan iSa MaltipUcation, definition of aS; »n-NairtzI 5, 16, j8, 34, 36, 44, +7, 30^, 310, 383 Nafliaddln at-TusI ig Nestelmuin, G. H> F. 1S7, 193 Nicomachus iiti, 119, 131, 179, 180, tSi, 1811, 183, 384, 185, 186, 187, 388, 189, 190, »9i, 191, «93, J94, 300, 363, 4»5 Nlion, K. C. j. 16 Number: debned by Thales, Eudoxus, Modei^tus, Aristotle, EncUd 180: Nico- macbuB and lamblicbus on^ 180: repre- sented by lidcs 1S8, and by points or dots 988-9 Oblong (of number): in Plato citbet 7^^1)^171 or ^ftp.-fi:nit 388 : but these terms denote two distinct divisions of plane numbers in Nicomachus, Theon of Smyrru and lam- blichos 189-90 Octahedron 98 Odd (number) : dcfe- of in Nicomachus 18 1 : Pjrtbagoreaii definition iSi : def. of odd and even by one another onscientific (Aristotle) 181 : Nioom. and Iambi, dis- tinguish three cbuses of odd numbers (i) prime and inoomposite, (a) secondary and composite, as eitremes, (3) secondary and composite in itself but prime and in- composite to one another, which ts inter- mediate 387 Odd-iiiMi nvM (number) : definition in Eucl. spurious 183-4, and dilTers from definitions by Nicomachus etc- itid^ Odd-iirrus odd (number): defined iti Enct. but not in Nioom- and lambL 1S4: Theon of Smyrna applies term to prime numbers 184 Oenopides of Chios 1 1 1 " Ordered" proportion {rtrvfitirtt dpoAryia), Interpolated deftnition of, 137 Pappos: lemma 00 Apollonjus' PioHt rtisia O4-5: problem from same work 81: assumes case of VI. 3 where CKtemal angle bisected (Simson's vi. Prop- A) 197: theorem from Apollonius' Plane Loci 198; theorem that ratio compounded of ratios of sides is equal to ralio of rectangles contained by sides ijo: +. >7. »9. *7. 7* 81, 113, 133, 3H, 1501 iji, i9> " Fturallelepipedal" (solid) numbers: two of the three factors differ by unity (Nicoma- ehus) 190 Peletarius (Pe)etiei) : on angle ^ itntaci and angle of semicircle 41: 47, 56^ 84, 146, 190 I'entflgon : decomposition of r^nlar pentagon into 30 elementary triangles 98: relation to pentagram 99 Pentagonal numbers 189 "Perftct" (of a class of numbers) 193-41 411-6: Pythagoreans applied term to tov 194: 3 also called "perfect" 194 Ptritirbtd pnpniitn (rtritpityiiitti dwiAffyfa) 136. 170-7 Pfleideier, (J. F- 1 Philoponus 134, i8i Plane numbers, prtiduct of two factors ("sides" or "length" and "breadth") 18 7-8; in Plato either square or oblong 187-8: similar plane numbers 193; one mean proportiorial between similar plane numb«s 371-1 Plato: construction of r^uUr solids Irom triangles 97-8: op^ifm iictiim 99: 7/5 as approximation to ^1, 119: on s<]uare and oolong numbers 188, 193 : on iw&fiMit (sqaare roots or surdi) 1881 190: theorem that between square numbers one mean suffices, between cube numbers two means necessary 194, 364 Playfair, John 1 Plutarch 98, 154 Fariim (corollary) to proposition prccedei "Q.E.D," or "Q.i.r." 8, 64: Porism to iv, tj mentioned by Proclus 109: Porism to VI. 19, 134 Polygonal numbers 189 Powers, R. E- 4»6 Prestet, Jean 416 Prime (number) I definitions of, 184-5: Aris- totle on two senses of "prime" i8j; t admitted as prime by Eucl- and Aristotle, but excluded by Nicomachus, Theon of Smyrna and laroblichus, who make prime a subdivision of add 184-3 = "prime and incomposite {da^vdtroi) " 184 : different names for prime, " odd-times odd" (Theon), •'linear" (Theon), "rectilinear'' (Thy- maridas), "euthymetric" (lamblichns) 185: prime absolutely or in themselves as dis- tinct from prime to one another (Theon) 18 J : definitioosof "prime to one another 18S-6 Proclus: on absence of formal divisions of proposition In certa^ cases, c-g. IV. 10, too: on use of "quindecagon for as* tronomy ill : +,39, 40, 193, 147, 169 Proportion : complete theory applicable to incommensurables as well as commen- Eurables is due to Eudoxus iii: old (Pythagorean) theory practically repre- sented by arithmetical theory of Eucl. vii. 113. in giving older theory as well Euclid simply followed tradition 113: Aristotle on general proof (new in his time) of theorem {aU^rmmda) in proportion j 13 : X- 5 as connecting two tnetmes 113: De Morgan on extension of meaning ^ nUi* ENGLISH INDEX 4ii to cover i]Qcomm«ii$urAbLe& i j 8 : pawei of txprasing incom mensurable ratio is power of ipproiimsitioii without limit up: in- terpolAted definitions of proportion as "wmeoess" or "similarity of ratics" (19 : definition in V. Def. 5 substituted for that of VII* Def. 3o because latter found Inade- qttace, not vke versa i-iW De Morgan's defence of V. Efef. 5 as necessary and sulGdent 11 1-^4 x v, Def» 5 corresponds to Weieratiass' conception of number in general and to Dedekind^s theory of ir- ratiomds 114-fi : alternatives for v. Def. 5 ^ a geometer- friend of Saccberi, fay Faifofer^ Ingram!^ Veronese^ Enriques and Amaldi 116: proportionals of Vll. Def. 30 {numbers) a particular case of those of v. Def- 5 (Sitnson^s Props. C» D and notes) 116-9 ■ proportion in three terms (Aristotle makes it four) the "least'' IJi : "con- tinuous " proportion (irivex^ or gvvyiii^vyt draXayltmn Euclid J{^ iiii\ir^\ 131, i^j: three *' proportions ^' 391^ but proportion far txteiUme or primary is continuous Or geometric 191-3; "discrete" or "dis- joined" (it^^ii^ri), iit^iv^iJifii) iji, igj; *'ordned" proportion {-rtTay^rn^ inter- polated definition of, 1 37 : " perturbed " proportitm (rtra^^^nj) 156^ 176-7 : ex- tettsive use of proportions in Greek geometry 1 87 : proporlions enable any quadratic equation with real roots to he solved JS7: supposed use of propositions of Book V. io arithmetical Books 314, 310 Fsellus ^ Ptolemy, Claudius: lemma about quadri- lateral in circle (Simson's vi. Prop. D) 145-7 - ^I't i'7t i'9 Pyiamidal numbers 190: pyramids truncated, twice-truncated etc. 191 Pythagoras : reputed discoverer of construc- tion of five regular solids 97 : introduced " the most per^t proportion in four terms and specially called ^harmonic'" into Greece 111: construction of figure equal to one and similar to another rectilineal ^ure 1J4 Pythagoreans : construction of dodecahedron in sphere 97 : construction of isosceles tiiaogle of IV. 10 and of r^Iar pentagon due to, 97-S : possible method of discovery of latter 97-9 : theorem about only three regular polygons fillin|r space round a point ^ : distinguished three sort* of in^artJ, arithmetic, geometric, harmonic lit' had theory of proportion applicable to conimensurables only ill ; 7/$ as ap- proximation to ^3, itq: definitions of 0011179: of even and odd 181: called lo " perfect" 194 Quadratic eqtuUoDi: tolution by meini of proportions j8j, i6}-Ji »66-7 : iapa/iit or condition of possibility of solving equation oF Ettd. VI. iS^ijj; one solution only given, for obvious reasons 160, 4641 167 : but method gives both roots if real 158 : exact correspondence of geometrical to algebraical solution 163-4, i6^7 Quadrilateral ; inscribing in circle of quadri- lateral equiangular to another 91-1: con- dition for inscribing circle in, 93, 95 : quadrilateral in cir^e, Ptolemy's lemma on (Simsmi's vi. Prop. D), 1)5-7 = quadri- lateral not a ' ' polygon " 139 " Quindecagon " (fifteen- angled figure): use- ful for astronomy 1 1 1 Radius: no Greek word for, 1 Ramus, P. iji Ratio : definition of, 1 16-9, no sufficient ground for r^ardtr^ it as spurious ir7, Barrow's defence of it 117 : method of transition from arithmetical to more general sense covering ineommensurables ti8: means of ix^nssing ratio of ineommen- surables is by approximation to any d^re« of accuracy 119: def. of grater ratio only <?it^ criterion (there are others) 130'; tests for greater equal and less ratios mutually exclusive ijo-i : test for greater ratio easier to apply than that for equal ratio 1 1^30 : a^uments about greater and less ratios unsafe unless they go back to original definitions (Simson on v- 10) 1^6-7 : rvm- peund T^iia 131-3, 189-90, 134: operation of compounding ratios 134: "ratio com- pounded of their sides" (careless exptes- sion) 14S: duplicate, trifluai€ etc. ratio as distinct from thuilt, tripU etc. 133: aliemate ratio, alUnuauU 134 : invtrs4 ratio, inverseiy 134 ; cempojition of ratio, (imferundt, different from c^mpaatdiHg ratios i 34- j : stparaiioH of ratio, stparaitdf (commonly dividend^ '35: effnverJMM of ratio, c&HvertittdQ 1 35 : ratio ex aefuaJi 136, ex atquaii in perturbed proporiioH 136 ! division of ratios used in Data as general method alleroative to compounding 14Q-JO ; names for particuJar arithmetical ratios 191 Rtfipr0cat or redprocedly reload figures : definition spurious 1S9 Riductio ad aistirdam, the only possible method of pioving iit. i, 8 "Rule of three": vi. 11 equivajeat to, 11 j Saccberi, Getolamo (16, 130 : proof of ex- istence of fourth proportional by vi. i, 1, 11, 170 Savile, H. 190 Sealent, a class of solid numbers 190 Scholia: iv. No. % ascribes Book iv. to Pythagoreans 97 : v. No. i attributes Book V. to Eudoxus iti Scholiast to Cltuds of Aristophanes 99 Stcth eanena attributed to Euclid 195 Stitor (of circle) : explanation of name : two kinds (i) iritb vertex at centre, (1) with vertex at drramference j ^ ENGLISH INDEX Secter-liii {figure} 5 : bisection of sucli a figure by straight line s S«ESii>fr, P. 416 St^ment of circle : angle ff/^ 4 : simitar u^- oients ; Semidrcle : angle cf, 4^ 39-4 1 ('^ Angle) ** (tngle in semicircie a right ^ngle, pre- Euclidesn proof 63 Separation of ratio, Stttipeffit XAtou, and leparando {AieAArrt} 1 35 : separandff and (omptntnds used relatively to one Hnather, not to oiigintil latio ifiS, 17a ScTTUS, C* 416 *SlM^f of plane and soHd numbers 187-8 Similar plane and solid numbers 993 : one mean between two similar plane numbers 37i-l» two means between two similar solid numbers 394, 373-5 Similar rectilineal fibres: def^ of^ given in Aristotle 1 88 : defTgives at once too little and loo much iSS: similar figures on straight lines which are proportional are themselve:s proportional and conversely (VI. n), alternatives for proposition 141-7 Similar sqrnvent^ of circles 5 Simon, Max J14, i^ Simpson, Thomas iii Simjon, R.: Props. C, D (Book v.) connect- ing proportioivals of vii. Def 10 as par- ticular case with those of V- Def^ A> 116-9: Anioms to Book v. tjj: Prop. B (inver- uon) 144: Prop. E \€ont^srisndt>) 175: shortens V, 8 by compressing tv^o cases into on« 1 5^-3 : important note showing daw in V. lo and giving alternative 156-7: Book vt. Prop. A extenJing VI. 3 to case where external angle bisected 197 ; Props. B, C, D 911-7 '• remarks on VI. i7-9i »S8-9 '■ '. 3. 8, ", 13 1 33t 34. ST. +J. 49. 53. 7«. 73. 79. 9=. "7, 13", 13'. 140. "43-4. MS. 146. 148. IS4. 'fii. >fil. '*3. 105, I7»^, 177, 179. rSo, (Bl, 183, 184, 185, (86, 1S9, r93, 19s, 109, III, 111, i3»-t. 138. »S». ?69» *70, 171-3 Site^ proper translation of rf^Xur^r^r in V. Def. J, 116-7, 189-90 Smith and Bryant, altemarive proofs 01 v. 16, ■7, 18 by means of vi. i, where magnitudes are straight lines or rectilineal areas i6j-6, "*9. 173-4 , , Solid numbers, three varieties accordiivg to relative length of sides loo-i Spktrited number, a particular kind of cube number 191 Square number, product of equal numbers 1S9, 191 : one mean between square numbera 194, 365-4 Stobaeus 180 Siibduplicale of iiny ratio found by vt. 13, 116 Swinden, J. H. van r8S Sylvester, J. 416 Tacquet, A. ill, ij8 Tannery, P. jm, (13 Tartaglia, Niccolb 1, 47 Taylor, H. M. t6, u, 19, 56, 75, 101, 117, i44i »47. »7' Tetrahedron 98 Thales 111, 180 Theodosius 37 Theon of Alexandria: interpolation in v. 13 and Porism 144 ; interpolated Porism to VI. 10, 139: additions to vi. 33 (about ttcton) 174-6: 4J, 109, 117, 119, 149, 151, 161, 186, 190, 134, 135, 140, 141, 156, 161, 3tt, jii, 411 Theon of Smyrna: 111, 119, 179, iSo, 181, 1S4, 185, 1S6, 188, 189, 19a, 191, 191, 193, '94, 41s Thrasyllus 191 Thymaridas 179, 1S5 Timaats of Plato 97-8, 194-5, 3^3 Todhuntcr, I., 3, 7, 11, 49. 51, 51. 67, 73, 90, 99, i7». 19s. lo'. "4, 108, 139, 171, 171, 300 Ttapeiium: name applied (o truncated pyramidal numbers (Theon of Smyrna) 19 1 Triangle : Heron's proof of expression (or area in terms of sides,s'i (i - o) (j - *) (j - f ) 87-8 : right-angled triangle which is half of equilateral triangle used lor construction of tetrahedron, octahedron and icosahedron (TtmofHi of Plato) 98 Triangular numbers ^89 Triplitate, distinct from iriplt, ratio i^i at-TQsl, sa Naslraddin Unit: definitions of, by ThymatidaSi "some Pythagoreans," Chrysippus, Aristotle and others 179: Euclid's definition was that of the " more recent " writers 179 : tiiv^-f connected etymological !y by Theon of Smyrna and Nicomachus with tihvoi (soli- tary) or fuir-li (rest) 179 Veronese, G. 30, 116 Vieta : on angle if eontatt 41 Walker 104, loS, 159 Wallis.John: on af>f/e ^ MJiAwf (" degree of curvature *') 41 Weierstrass 114 Woepcke 5 ^ , Zenodorus 176 ,, .^,